Forced convectionIn this case, the heat transfer coefficient h depends on the flow velocity u, characteristic length scale L, fluid conductivity k, viscosity μ, specific heat capacity Cp and density ρ. These parameters and their dimensions are listed in Table. Using the fundamental dimensions of M for mass, L for length, T for time and θ for temperature

Table: Variables and dimensions for forced convectionParameter Dimensions

h MT-3 θ-1 u LT-1

L Lk MLT-3 θ-1

μ ML-1 T-1

C p L2T-2 θ-1

ρ ML -3

We start then by writing the functional relationship as;

h= f ( u,L,k,Cp,μ,ρ )

The number of variables is 7 and the number of dimensions is four, so we will be able to get three non-dimensional parameters. We choose a set of repeated variables containing the four dimensions such that they do not form a π group on their own. By inspection, we can see that the variables k, L, μ and ρ cannot form a π because only the first one contains the dimension θ.

We then use this set of variables with one of the remaining variables one at a time to extract the non-dimensional groupings.

ka, Lb, μc, ρd h = (MLT-3 θ-1)a ( L)b ( ML-1 T-1)c (ML -3)d ( MT-3 θ-1 ) = π1 M0L0T0θ0

We can then create a system of linear equations to compute the values of the exponents a,b,c and d that lead to a non-dimensional group. So:

For M: a + c + d + 1 = 0For T: - 3a - c - 3= 0For θ : - a - 1 = 0For L: a + b - c - 3d = 0

Solving these equations simultaneously gives:a= -1 b= 1 c= 0 d= 0

This leads to the following non-dimensional group k-1 Lh. Thus the first non-dimensional group is:

π1 = h L/k

Repeating the same process with the variable u

ka Lb μc ρd u = (MLT-3 θ-1)a ( L )b ( ML-1 T-1)c (ML -3)d (LT-1) = π2 M0L0T0θ0

For M a + c + d = 0For L a+ b –c -3d +1 = 0For T -3a –c -1 = 0For θ -a = 0Solving these equations simultaneously gives

a = 0 b =1 c = -1 d =1

This results in the following non-dimensional group:

π2 = ρu L / μ

Repeating the procedure using the variable Cp leads to a third non-dimensional group of the form

π3 = μ Cp / kUsing the above non-dimensional groups, the functional relation in Equation h= f ( u,L,k,Cp,μ,ρ ) can be expressed as:

h L/k = f { ρu L / μ , μ Cp / k}

We will now give a physical interpretation of the three non-dimensional groups derived above

h L/k : This group is called the Nusselt number Nu. This represents the dimensionless heat transfer coefficient and can be thought of as the ratio of the heat transfer by convection to that of conduction through the fluid.

q convq cond

= h∆ Tk ∆ T /L

=hLk

=Nusselt No. Nu

Nu = 1 pure conduction.

A value of Nusselt number around 1 implies either convective effect is weak or not present. The local Nusselt number is usually termed Nux. If the average value is used over a surface, then the term Nu L is used.ρ u L / μ: This group is known as the Reynolds number Re. It represents the ratio of the inertia forces to the viscous forces in the fluid.

ℜ= Inertia forcesViscous forces

= ρu Lμ

Its value can give an indication of the state of the boundary layer and whether the flow is

laminar, turbulent or in transition. At large Reynolds numbers (turbulent flow) the inertia forces are large relative to the viscous forces. At small or moderate Reynolds numbers (laminar flow), the viscous forces are large enough to suppress these fluctuations and to keep the fluid “inline.”

Again here the local Reynolds number is termed Rex and the Reynolds number based on the length scale of the flow domain is termed ReL.

μ Cp / k: The Prandtl number Pr. This can be written as follows

Pr = μ Cp / k = (μ/ρ)( k/ ρ Cp) = ν/α

Pr= Molecular diffusivity of momentumMolecular diffusivity of heat

= να

=μCpk

From which, we can se the Pr is the ratio of the momentum diffusivity (ν) to the thermal Diffusivity (α) It provides a measure of the relative effectiveness of the transport by diffusion of momentum and energy in the velocity and the thermal boundary layers respectively. Heat diffuses very quickly in liquid metals (Pr«1) and very slowly in oils (Pr»1) relative to momentum.

For gasses Pr~1.0 and in this case momentum and energy transfer by diffusion are comparable. In liquid metals Pr << 1 and the energy diffusion rate is much greater than the momentum diffusion rate. For oils Pr >> 1 and the opposite is true.

From the above interpretation it follows that the Pr number affects the growth of the velocity and the thermal boundary layers, i.e. in laminar flow

Thermal boundary layer thickness

= Prn ( n is a positive number) Velocity boundary layer thickness

Equation h L/k = f { ρu L / μ , μ Cp / k} forms the basis for general formulations for the non-dimensional heat transfer coefficient as a function of the Reynolds number and the Prandtl numbers as follows:

Nu α (Re)a (Pr)b or

Nu = C (Re)a (Pr)b

The constants C, a and b can be obtained either by experiments, numerical methods or from analytic solutions if they were possible. In the following subsection we will present the values of these coefficients for common geometries encountered in engineering applications

EXTERNAL FORCED CONVECTIONLaminar flow over a flat plate

In the simple case of an isothermal flat plate, as mentioned earlier, analytical methods can be used to solve either the Navier-Stokes equations or the boundary layer approximation for Nux. This then can be integrated to obtain an overall Nusselt number NuL for the total heat transfer from the plate. The derivation of the analytic solution for the boundary layer equations can be found in Long (1999). This leads to the following solutions:

hx x/k = 0.332 [ μ Cp / k ]1/3 [ ρ u x / μ ]1/2

Nux = 0.332 (Pr)1/3 ( Rex )1/2

The above physical properties (k, c p , ρ , μ ) of the fluid can vary with temperature and are therefore

evaluated at the mean film temperature T film = T s+T ∞2

Where T s is the plate is surface temperature and T is the temperature of the fluid in the far stream.

Above equation is applicable for Rex < 5x105 i.e. laminar flow and Pr ≥ 0.6 (air and water included).In most engineering calculations, an overall value of the heat transfer coefficient is required rather than the local value. This can be obtained by integrating the heat transfer coefficient over the plate length as follows:

qx = (area of element ) hx (Ts – T ) =bdx(Ts – T )

And for the plate entire length

Q= ∫ q x =bhL(Ts – T )

Where hL is the average convective coefficient and Q is the total heat transfer from the plate. If we substitute for qx from Equation qx = (area of element) hx (Ts – T ) = bdx(Ts – T ) and rearrange we have,

hL = 1hx dxL Type equation here. hx = L h L

dx

If we now substitute for h x from Equation

hx xk = 0.332 [ μ Cp

k ]1/3 [ ρ u xμ ]1/2

and integrate the resulting equation the average heat transfer coefficient is

hL Lk = 0.664 Pr1/3 ReL

1/2 or

NuL = 0.664 Pr1/3 ReL1/2

This is valid for ReL < 5x105

Comparing the above equation with Nux = 0.332 (Pr)1/3 ( Rex )1/2

hL = 2hat x=L

Turbulent flow over a flat plate

Transition from laminar to turbulent flow over a flat plate generally takes place at a Reynolds number of approximately 5x10 5. This value however can vary up to an order of magnitude either way dependent on the state of free stream turbulence and the smoothness of the plate.Figure 3.8 shows a schematic of the velocity profile near the wall for turbulent and laminar flow

It is apparent that the velocity gradient is much steeper for turbulent flow.

For the same temperature difference between the wall and the free stream flow, turbulent flow will produce a larger heat transfer coefficient. Physically, this can be explained by the fact that there is more mixing of the flow due to turbulent fluctuations leading to a higher heat transfer.

NuL = [0.664 Rex,c0.5 + 0.037( ReL

0.8- Rex,c0.8)] Pr 1/3

Where Rex,c is the Reynolds number at which transition occurs. Assuming that transition occurs.

At ReL = 5x105 ,

NuL = 0.037( ReL0.8- 871)] Pr 1/3

If the length at which transition occurs is much smaller than the total length L of the plate, then above equation can be approximated by

NuL = 0.037ReL0.8 Pr 1/3

External Flow Across Cylinders And Spheres

Flow across cylinders and spheres are frequently encountered in practice. For example, the tubes in a shell-and-tube heat exchanger involve both internal flow through the tubes and external flow over the tubes, and both flows must be considered in the analysis of the heat exchanger. Also, many sports such as soccer, tennis, and golf involve flow over spherical balls.

The characteristic length for a circular cylinder or sphere is taken to be the external diameter D. Thus, the Reynolds number is defined as

Re =ρDu/μ Where u is the uniform velocity of the fluid as it approaches the cylinder or sphere. The critical Reynolds number for flow across a circular cylinder sphere is about Re ≅ 2 x 105. That is, the boundary layer remains laminar for about Re ≤ 2 x 105 and becomes turbulent for Re ≥ 2 x105.

Cross flow over a cylinder exhibits complex flow patterns, as shown in Figure. The fluid approaching the cylinder branches out and encircles the cylinder, forming a boundary layer that wraps around the cylinder. The fluid particles on the mid plane strike the cylinder at the stagnation point, bringing the fluid to a complete stop and thus raising the pressure at that point. The pressure decreases in the flow direction while the fluid velocity increases.

Stagnation point: Location of zero velocity and maximum pressure. Followed by boundary layer development under a favorable pressure gradient and hence acceleration of the free stream flow.

Location of separation depends on boundary layer transition. Since the momentum of fluid in a turbulent boundary layer is larger than in the laminar boundary layer, it is reasonable to expect transition to delay the occurrence of separation.

At very low upstream velocities (Re ≤1), the fluid completely wraps around the cylinder and the two arms of the fluid meet on the rear side of the cylinder in an orderly manner. Thus, the fluid follows the curvature of the cylinder.

At higher velocities, the fluid still hugs the cylinder on the frontal side, but it is too fast to remains attached to the surface as it approaches the top of the cylinder. As a result, the boundary layer detaches from the surface, forming a separation region behind the cylinder. Flow in the wake region is characterized by random vortex formation and pressures much lower than the stagnation point pressure.

– For Reynolds number > 200,000 on a smooth cylinder, the boundary becomes turbulent. Flow separation is delayed, and the wake is small then that when the boundary is laminar.

Flow separation occurs at about θ≈ 80o (measured from the stagnation point) when the boundary layer is laminar and at about θ≈ 140o when it is turbulent.

Effect of Surface RoughnessThe surface roughness, in general, increases the drag coefficient in turbulent flow. This is especially the case for streamlined bodies.For blunt bodies such as a circular cylinder or sphere, however, an increase in the surface roughness may actually decrease the drag coefficient for a sphere. This is done by tripping the flow into turbulence at a lower Reynolds number, and thus causing the fluid to close in behind the body, narrowing the wake and reducing pressure drag considerably. This result in a much smaller drag coefficient and thus drag force for a rough surfaced cylinder or sphere in a certain range of Reynolds number compared to a smooth one of identical size at the same velocity. At Re = 105, for example, CD = 0.1 for a rough sphere with ε/D = 0.0015, whereas CD =0.5 for a smooth one. Therefore, the drag coefficient in this case is reduced by a factor of 5 by simply roughening the surface. Note, however, that at Re =106, CD = 0.4 for the rough sphere while CD =0.1 for the smooth one. Obviously, roughening the sphere in this case will increase the drag by a factor of 4.

The discussion above shows that roughening the surface can be used to great advantage in reducing drag, but it can also backfire on us if we are not careful specifically, if we do not operate in the right range of Reynolds number.With this consideration, golf balls are intentionally roughened to induce turbulence at a lower Reynolds number to take advantage of the sharp drop in the drag coefficient at the onset of turbulence in the boundary layer (the typical velocity range of golf balls is 15 to 150 m/s, and the Reynolds number is less than 4 x105). The critical Reynolds number of dimpled golf balls is about 4 x 104. The occurrence of turbulent flow at this Reynolds number reduces the drag coefficient of a golf ball by half. For a given hit, this means a longer distance for the ball.

Heat Transfer CoefficientFlows across cylinders and spheres, in general, involve flow separation, which is difficult to handle analytically. Therefore, such flows must be studied experimentally or numerically.Generally the overall average Nusselt number for heat transfer with the entire object is important. As with a flat plate, correlations developed from experimental data to compute Nu as a f(Rem,Prn)

Overall Average Nusselt number

NuD= h̄ Dk

= C ReDm Pr1/3

All properties are evaluated at the bulk mean temperature.

For circular cylinders the value of C and m are as under

Range of Reynolds Number Nusselt number0.4–4 Nu =0.989Re0.330 Pr1/3

4–40 Nu = 0.911Re0.385 Pr1/3

40–4000 Nu = 0.683Re0.466 Pr1/3

4000–40,000 Nu = 0.193Re0.618 Pr1/3

40,000–400,000 Nu = 0.027Re0.805 Pr1/3

INTERNAL FORCED CONVECTION

Laminar Flow In Pipes

If the Reynolds number Red, based on the pipe diameter is less than about 2300, then the flow laminar. For Reynolds number above 2300 the boundary layer developing at the entrance of the pipe undergoes transition and becomes turbulent leading to a fully developed turbulent flow

In pipe flow, we seek to determine a heat transfer coefficient such that Newton’s law of coolingis formulated as:

q = h( T s - T m)

Where T sis the pipe-wall temperature and T m is the mean temperature in the fully developed profile in the pipe. The mean fluid temperature is used instead of the free stream temperature for external flow.In a similar fashion to the average Nusselt number for a flat plate, we define the average Nusselt number for pipe flow as:

Nu d =hd / k, where d is the pipe diameter.

From the derivation of formulae for the average Nusselt number for laminar follow resulting from the solution of the flow equations, it turns out the the Nusselt number is constant, and does not depend on either the Reynolds or Prandtl numbers so long as the Reynolds number is below2300.However, two different solutions are found depending on the physical situation. For constant heat flux pipe flow, the Nusselt number is given by

Nud = 4.36

While for a pipe with constant wall temperature, it is given by:

Nud =3.66

Note that to determine h from above equations, the heat transfer coefficient needs to be determined at the mean temperature T m. For pipes where there is a significant variation of temperature between entry and exit of the pipe (such as in heat exchangers), then the fluid properties need to be determined at the arithmetic mean temperature between entry and exit

i.e. at T mi−T me

2

Turbulent Flow In Pipes

Determination of the heat transfer coefficient for turbulent pipe flow analytically is much more involved than that for laminar flow. Hence, greater emphasis is usually placed on empirical correlations.

The classic expression for local Nusselt number in turbulent pipe flow is given by:

Nud = 0.023 (Red )4/5Pr2/3

However, it is found that the Dittus-Boelter equation below provides a better correlation with measured data:

Nud = 0.023 (Red )4/5Prn

where n = 0.4 for fluid is heating (Ts > Tm) and n = 0.3 for fluid is cooling (Ts < Tm)

The above equation is valid for:

Red > 104, and L / D > 10

T s - T f < 5 C for liquids and T s - T f < 55 C for gases

For larger temperature differences use of the following formula is recommended

Nud = 0.027 (Red )4/5Pr2/3 [ μμs ]

For 0.7 ≤Pr ≤ 16700,

Re d ≥ 10,000 and L/ d ≥10

Where µs is the viscosity evaluated at the pipe surface temperature. The rest of the parameters are evaluated at the mean temperature.

Flow Through Tube Annulus

Some simple heat transfer equipments consist of two concentric tubes, and are properly called double-tube heat exchangers. In such devices, one fluid flows through the tube while the other flows through the annular space. The governing differential equations for both flows are identical. Therefore, steady laminar flow through an annulus can be studied analytically by using suitable boundary conditions.

If the channel through which the fluid flows is not of circular cross section, it is recommended that the heat-transfer correlations be based on the hydraulic diameter DH, defined by

DH = 4 AP

where A is the cross-sectional area of the flow and P is the wetted perimeter. This particular grouping of terms is used because it yields the value of the physical diameter when applied to a circular cross section. The hydraulic diameter should be used in calculating the Nusselt and Reynolds numbers

Consider a concentric annulus of inner diameter Di and outer diameter Do. The hydraulic diameter of annulus is

DH = 4 AP = 4 π Do

2 – Di2 / π (Do + Di ) = Do – Di

When Nusselt numbers are known, the convection coefficients for the inner and the outer surfaces are determined from

Nui = hi DH / k Nuo = ho DH / k

Natural convection

An example of heat transfer by natural convection is that resulting from the external surface of radiators of a central heating system or an electric heating element. In this case, as the surrounding fluid is heated, its density reduces. This results in this fluid rising and will be replaced by colder fluid from the surrounding resulting in a circulation loop as shown in Figure

Generally speaking, natural convection velocities are much smaller than those associated withforced convection resulting in smaller heat transfer coefficients.If ρ∞ is the density of the “undisturbed” cold fluid and ρ is the density of warmer fluid then the buoyancy force per unit volume F of fluid is:

F = (ρ∞ - ρ )g ………….. 1

Where g is the acceleration due to gravity

The variation of density with temperature is:

ρ∞ = ρ ( 1 + β ∆ T) ……………. 2

where β is the volumetric thermal expansion coefficient (1/ K ) and ∆T is the temperature difference

between the two fluid regions. If we substitute for ρ∞ from Equation 2 intoEquation 1 then the buoyancy force per unit volume of fluid is given a

F ={ ρ ( 1 + β ∆ T) – ρ} g or

F = ρ β g ∆ T

Therefore, in the case of natural convection, h could depend on a characteristic length L. A temperature difference ∆T, the conductivity k, the viscosity μ the specific heat capacity c p , the density ρ, and the volumetric thermal expansion coefficient β of the fluid. β is usually grouped with g and ∆T as one term ( βg∆T ) as this group is proportional to the buoyancy force.

We will now use the principles of dimensional analysis to work out a set of non-dimensional parameters to group the parameters affecting natural convection. Thus

h = f ( L, k,μ, Cp , ( βg∆T ), ρ )

These have the dimensions as shown in Table

Table: Parameters affecting natural convection and their dimensions

h MT-3θ-1

βg∆T L T-2θL L∆T θk ML T-3θ-1

μ ML-1 T-1

c p L2 T-2θ-1

ρ ML-3

Again there are seven parameters and four dimensions which should lead to three non dimensional groups.

Selecting the following repeated variables: k, L, μ and ρ as they cannot form a non dimensional group because only k has the dimension of temperature. We will then use the remaining variables as repeated variables one at a time. If we start with h, we get the following:

k aLb μc ρd h = ( ML T-3θ-1)a (L)b (ML-1 T-1)c (ML-3)d (MT-3θ-1) = M0 T0 L0 θ0

which leads to the following non-dimensional group

π1 = hL/k

If we choose βg∆T , we get:

k aLb μc ρd βg∆T = ( ML T-3θ-1)a (L)b (ML-1 T-1)c (ML-3)d (L T-2θ) = M0 T0 L0 θ0

For M: a + c + d = 0For T: - 3a - c - 2 = 0For θ: - a = 0

For L: a + b - c - 3d + 1 = 0

Solving these equations simultaneously gives:

a = 0 , b = 3 , c = -2 , d = 2

This results in the following non-dimensional group:

π2 = ρ2βg∆TL3 / μ2

Repeating the procedure using the variable Cp leads to a third non-dimensional group :

π3 = μ Cp / k

Using the above non-dimensional groups, the functional relation can be expressed as:

h Lk = f { ρ 2 βg ∆ TL 3 , μ C p }

μ2 k

or Nu = f [ Gr, Pr ]

where Gr = ρ2βg∆TL3 / μ2 is the Grashof number.

This dimensionless group is the ratio of the buoyancy forces to the square of the viscous forces in the fluid. Its role in natural convection is similar to the role of the Reynolds number in forced convection. At high Gr numbers the buoyancy forces are large compared to the viscous forces which tend to hold the fluid particles together and thus convection can occur.

In natural convection, we are also likely to encounter what is known as the Rayleigh number which is the product of the Grashof number and the Prandtl number:

Ra = Gr Pr = βg∆TL3 / vα

This is the ratio of the thermal energy liberated by buoyancy to the energy dissipated by heat conduction and viscous drag.It is customary to use the following expression for the Nusselt number in free convection, in a similar fashion to that used in forced convection:

Ra = C Gra Prb

where C, a and b are constants that can be determined either analytically experimentally, or using computational methods. Traditionally, experimental were methods used, but numerical procedures are

becoming increasingly more used in recent years. Analytical methods are only possible for a limited number of very simple cases.

In all cases, the fluid properties should be evaluated at T film = (T s + T ∞ ¿ / 2 for gasses β= 1 / T film

.