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In this article we will discuss about:- 1. Meaning of Convection 2. Laminar and Turbulent Flow of Fluid Affecting Convection 3. Newton-Rikhman Law 4. Dimensionless Group 5. Significance.

**Contents:**

- Meaning of Convection
- Laminar and Turbulent Flow of Fluid Affecting Convection
- Newton-Rikhman Law: Convection Rate Equation
- Dimensionless Group for Convection
- Significance of Dimensionless Groups for Convection

#### 1. Meaning of Convection:

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Thermal convection occurs when a temperature difference exists between a solid surface and a fluid flowing past it. Convection is essentially a process of energy transport affected by the circulation or mixing of a fluid medium which may be a gas, a liquid or a powdery substance. The transport of heat energy during convection is directly linked with the transport of medium itself, and as such convection presents a combined problem of conduction, fluid flow and mixing.

**Free and Forced Convection: **

**With respect to the cause of fluid circulation or flow, two types of convection are distinguished: **

**i. Free Convection: **

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Circulation of bulk fluid motion is caused by changes in fluid density resulting from temperature gradients between the solid surface and the main mass of fluid. The stagnant layer of fluid in the immediate vicinity of the hot body gets thermal energy by conduction. The energy thus transferred serves to increase the temperature and internal energy of fluid particles. Because of temperature rise, these particles become less dense and hence lighter than the surrounding fluid particles.

The lighter fluid particles move upwards to a region of low temperature where they mix with and transfer a part of their energy to the cold particles. Simultaneously the cool heavier particles descend downwards to fill the space vacated by the warm fluid particles. The circulation pattern, upward movement of the warm fluid and downward movement of cool fluid, is called convection currents. These currents are setup naturally due to gravity alone and are responsible for heat convection.

Designers of furnaces, house heating systems, architectural projects, roads and concrete structures will be concerned with free convection. Since there are no density forces (no gravitational field) in the orbiting satellites, the space vehicles with a zero gravity trajectory, free convection would be non-existent in such vehicles.

**ii. Forced Convection: **

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Flow of fluid is caused by a pump, a fan or by the atmospheric winds. These mechanical devices provide a definite circuit for the circulating currents and that speeds up the heat transfer rate. Examples of forced convection are cooling of internal combustion engines, air conditioning installations and nuclear reactors, condenser tubes and other heat exchange equipment.

** 2. Laminar and Turbulent Flow of Fluid Affecting Convection: **

**The convection heat is affected to an appreciable extent by the nature of fluid flow. In the realms of fluid mechanics, essentially two types of fluid flow are characterised: **

**i. Laminar Flow: **

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The fluid particles move in flat or curved un-mixing layers or streams and follow a smooth continuous path. The paths of fluid movement are well-defined and the fluid particles retain their relative positions at successive cross-sections of the flow passage. There is no transverse displacement of fluid particles; the particles remain in an orderly sequence in each layer. Soldiers on a parade provide a somewhat crude analogy to laminar flow.

**ii. Turbulent Flow: **

The motion of fluid particles is irregular, and it proceeds along erratic and unpredictable paths. The stream lines are intertwined and they change in position from instant to instant. Fluctuating transverse velocity components are superimposed on the main flow, and the velocity of individual fluid elements fluctuate both along the direction of flow and perpendicular to it. Obviously a turbulent flow is eddying and sinuous rather than rectilinear in character. The turbulent flow resembles a crowd of commuters in a rail road station during the rush hour.

Osborne Reynolds, an English scientist, confirmed the existence of these two regimes experimentally and postulated that under certain conditions there could be transition from laminar to turbulent flow and vice versa.

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**His investigations revealed that the nature of fluid flow is governed by the following parameters: **

i. Mean flow velocity V

ii. Density of fluid p

iii. Dynamic viscosity of the fluid µ

iv. Characteristic dimension of the flow passage, for example the diameter d of the pipe.

A grouping of these variables results into a dimensionless quantity, R_{e} = Vdρ/µ, called the Reynolds number. This number represents the ratio of inertia to viscous forces. At low Reynolds number, the viscous forces predominate and the flow is laminar. At high values of Reynolds number, the inertia forces overcome the viscous friction forces and consequently the fluid layers break up into a turbulent flow.

For fluid flow through a pipe, low Reynolds number upto 2300 is indicative of laminar flow. From R_{e} = 2300 to 6000, the laminar flow begins a transition to turbulent flow. Usually the flow is completely turbulent at R_{e} = 6000.

In many flow situations, the duct is not circular but is rectangular, trapezoidal or even an annulus formed by a tube within another tube. In that case, the characteristics dimension d in the relation R_{e} = Vdρ/µ is the equivalent (hydraulic) diameter which is defined as four times the cross-sectional flow area divided by the wetted perimeter.

If the annulus (a flow passage formed by a tube within a tube) has an inner diameter (outer diameter of inner tube) of d_{1} and an outer diameter (inner diameter of outer tube) of d_{2} then the equivalent diameter is-

** 3. Newton-Rikhman Law: Convection Rate Equation: **

Regardless of the particular nature (free or forced), the appropriate rate equation for the convective heat transfer between a surface and an adjacent fluid is prescribed by Newton’s law of cooling.

Where, Q is the convective heat flow rate, A is the surface area exposed to heat transfer, t_{s} is the surface temperature of the solid, and t, is the undisturbed temperature of the fluid. The constant of proportionality h relates the heat transfer per unit time and unit area to the overall temperature difference. The unit of h are W/m^{2} deg, and it is referred to as convective heat transfer coefficient, the surface conductance or the film coefficient.

**The value of film coefficient is dependent upon: **

i. Surface conditions: roughness and cleanliness

ii. Geometry and orientation of the surface- plate, tube and cylinder placed vertically or horizontally

iii. Thermo-physical properties of the fluid- density, viscosity, specific heat, coefficient of expansion and thermal conductivity

iv. Nature of fluid flow- laminar or turbulent

v. Boundary layer configuration

vi. Prevailing thermal conditions

**Typical values of convective coefficient are:**

Convection mechanisms involving phase changes lead to important field of boiling (evaporation) and condensation. The convection coefficients for boiling and condensation lie in the range 2500 – 100,000 W/m^{2}K.

**Example 1: **

A motor cycle cylinder consists of ten fins, each 150 mm outside diameter and 75 mm inside diameter. The average fin temperature is 500°C and the surrounding air is at 20°C temperature. Make calculations for the rate of heat dissipation from the cylinder fins by convection when (i) motor cycle is stationery and convection coefficient h = 6 W/m^{2}K (ii) motor cycle is moving at 60 km/hr and h = 75 W/m^{2}K.

**Solution: **

**Since both sides of each fin are exposed to the surroundings air, the surface area for convective heat transfer is: **

**Example 2: **

Forced air flows over a convective heat exchanger in a room heater, resulting in a convective heat transfer coefficient 1.136 kW/m^{2}K. The surface temperature of heat exchanger may be considered constant at 65° C, and the air is at 20°C. Determine the heat exchanger surface area required for 8.8 kW of heating.

**Solution: **

The convective heat flow from a solid surface to the surrounding fluid is given by Newton’s law of cooling-

** **

**4. Dimensionless Group for Convection:**

**Dimensionless Group for Free Convection: **

The different variables specifying the system behaviour have been indicated in Fig. 9.1., which represents the free convection of fluid flow over a flat plate.

**The physical quantities with their dimensions is M-L-T-ϴ-H system of units are: **

The coefficient of thermal expansion, β, is prescribed by the relation;_{}

ρ_{1} = ρ_{2} {1 + β (t_{2} – t_{1})}

where ρ_{1} and p_{2} are the fluid densities at temperature t_{1} and f_{2} respectively,

Buoyant force = (ρ_{1} – ρ_{2})g = [ρ_{2} {1 + β (t_{2} – t_{1})} – ρ_{2}] g = ρ_{2} (δg Δt)

This suggests the inclusion of variables β, g, ΔT into the list of those important to the natural convection situation. The parameter (βg ΔT) represents the buoyant force and has the dimensions of (LT^{-2}).

**Buckingham’s π-Method: **

It can be premised that the functional relationship is,

*f* (π, ρ, k, c_{p}, βg ΔT, I, h) = 0

There are 8 physical quantities (βg and ΔT are counted separately) and 5 fundamental units; hence (8 – 5) or 3π-terms. We choose fluid viscosity π, thermal conductivity k, and product (βg Δt) and the characteristic length I as the core group (repeated variables) with unknown exponents, and establish the π terms as follow-

Equating the exponents of fundamental dimensions on both sides-

The above step has been worked on the postulate that square of a π-term is also non-dimensional.

Following the same procedure, one would obtain;

Where,

Nu = hl/k, a dimensionless group called Nusselt number

Gr = (l^{3} ρ^{2} βg ΔT/µ^{2}), a dimensionless group called Grashof number

Pr = µ c_{p}/k, a dimensionless group called Prandtl number

It is a usual practice to rewrite the above correlation in the form

Nu = C (Gr)^{a} (Pr)^{b}

The constant C and the exponents a and b are evaluated through experiments.

**Dimensionless Groups for Forced Convection:**

The different variables specifying the system behaviour have been indicated in Fig. 9.2 which represents the forced convection of fluid flow over a flat plate.

**The physical quantities with their dimensions in M-L-T-ϴ-H system of units are:**

**Buckingham’s n-Method:**

It can be premised that the functional relationship is

*f* (µ, ρ, k, c_{p} Δt, V, I, h) = 0

There are 8 physical quantities and 5 fundamental units, hence (8 – 5) or 3 π-terms. We choose fluid viscosity µ, thermal conductivity k, velocity V and the characteristic length l as the core group (repeated variables) with unknown exponents and establish the π-terms as follows-

**Equating the exponents of fundamental dimensions on both sides: **

Where,

Nu = (hl/k), a dimensionless group called Nusselt number

Re = (ρ Vl/µ), a dimensionless guoup called Reynoles number

Pr = (µ c_{p}/k), a dimensionless number called Prandtl nunber

It is usual practice to rewrite the above correlation in the form-

Nu = C (Re)^{a} (Pr)^{b}

The constant C and the exponents a and b are evaluated through experiments.

**Note: **

If µ, ρ, c_{p} and V were chosen as the core group (repeated variables), then the analysis would have yielded the following non-dimensional groups-

Where, St is called the Stanton number.

Accordingly another correlation for forced convection would be of the form-

St = ɸ (Re, Pr)

** 5. Significance of Dimensionless Groups for Convection: **

**i. Reynolds Number Re: **

Ratio of the interia force to the viscous force:

Reynolds number is indicative of the relative importance of inertial and viscous effects in a fluid motion. At low Reynolds number, the viscous effects dominate and the fluid motion is laminar. At high Reynolds number, the inertial effects lead to turbulent flow and the associated turbulence level dominates the momentum and energy flux.

**ii.** Grashof Number Gr indicates the relative strength of the buoyant to viscous forces. From its mathematical formulation,

Obviously the Grashof number represents the ratio of the product of buoyant and inertia forces to the square of the viscous forces. Grashof number has a role in free convection similar to that played by Reynolds number in forced convection.

**iii.** Prandtl Number Pr is indicative of the relative ability of the fluid to diffuse momentum and internal energy by molecular mechanisms. From its mathematical formulation,

Apparently Pr is the ratio of the kinematic viscosity to thermal diffusivity of the fluid. The kinematic viscosity indicates the momentum transport by molecular friction and thermal diffusivity represents the heat energy transport through conduction. Obviously Pr provides a measure of the relative effectiveness of momentum and energy transport by diffusion.

The Prandtl number is connecting link between the velocity field and the temperature field, and its value strongly influences relative growth of velocity and thermal boundary layers. Mathematically,

Where, δ and δ_{t} are the thickness of velocity and thermal boundary layers respectively, and n is a positive exponent. For oils δ_{t} << δ; for gases δ_{t} ≈ δ; and for liquid metals δ_{t} >> δ.

**iv. Nusselt Number: **

Consider a heated wall surface at temperature t_{s} over which a fluid is flowing with undisturbed velocity U_{∞} and temperature t_{∞}. The particles of fluid in intimate contact with the plate tend to adhere to it, and a region of variable velocity builds up between the plate surface and the free fluid stream as indicated in Fig. 9.3.

The fluid velocity decreases as it approaches the solid surface, reaching to zero (no slip condition) in the fluid layer immediately next to the surface. This thin layer of stagnated fluid has been called the hydrodynamic boundary layer. The quantity of heat transferred is highly dependent upon the fluid motion within this boundary layer, being determined chiefly by the thickness of the layer. The boundary layer thickness 5 is arbitrarily defined as the distance y from the plate surface at which the velocity approaches 99% of free stream velocity.

Likewise a region of fluid motion near the plate in which temperature gradients exist is the thermal boundary layer and its thickness δ_{f} is defined as the value of transverse distance y from the plate surface at which,

At the plate surface, there is no fluid motion and the energy transport can occur only by conduction. From energy balance, this heat transport must equal the heat transferred by convection into the rest of the fluid. Thus-

Heat flow rate is thus dependent upon temperature gradient at the wall, and the temperature gradient is influenced by the fluid velocity; high temperature gradients are associated with the higher velocities.

If temperature field of the fluid varies only in the direction of the coordinate normal to the plate surface, then-

Thus, the convective coefficient h can be evaluated from a knowledge of fluid temperature distribution in the neighbourhood of the surface.

Introducing a characteristic dimension l, the equation 9.6 can be recast as-

The dimensionless parameter hl/k is called Nusselt number. Apparently the Nusselt number may be interpreted as the ratio of temperature gradient at the surface to an overall or reference temperature gradient. The parameter-

represents the dimensionless slope of the temperature distribution curve at the surface.

The Nusselt number is a convenient measure of the convective heat transfer coefficient. For a given value of Nusselt number, the convective surface coefficient h is directly proportional to thermal conductivity k of the fluid, and inversely proportional to the significant length l.

**v.** Stanton Number St is the ratio of heat transfer coefficient to the flow of heat per unit temperature rise due to the velocity of the fluid.

**Thus the Stanton number can be expressed in terms of other dimensionless numbers as: **

It should be noted that Stanton number can be used only in correlating forced convection data. This becomes obvious when we observe the velocity V contained in the expression for Stanton number.

**Example 3****:**

**The temperature profile at a particular location in a thermal boundary layer is prescribed by an expression of the form: **

t(Y) = A – By + Cy^{2}

where, A, B and C are constants. Set up an expression for the corresponding heat transfer coefficient.

**Solution: **

The heat transfer coefficient is given by the expression

**Example 4: **

**The temperature profile at a particular location on the surface of plate is prescribed by the identities: **

If thermal conductivity of air is stated to be 0.03 W/m-deg, determine the value of convective heat transfer coefficient in each case.

**Solution: **

The convective heat transfer coefficient is prescribed by the relation-