Convection Heat Transfer Heat transfer: Heat transfer (or heat) is thermal energy in transit due to a temperature difference. According to the 2nd law of thermodynamics heat is transferred from a higher temperature body to a lower temperature body. Modes of heat transfer: (i) Conduction: Mechanism of heat transfer through a solid or fluid in the absence of any fluid motion. (ii) Convection: Mechanism of heat transfer through a fluid in the presence of bulk fluid motion. (iii) Radiation: The energy of the radiation field is transported by electromagnetic waves 9or alternatively, photons0. Radiation heat transfer does not require material medium. Types of convection heat transfer: Convection heat transfer depends on how the fluid motion is initiated. (i) Natural or free convection; (ii) Forced convection; Natural or free convection: In natural convection, any fluid motion is caused by natural means such as the buoyancy effect, which manifests itself as the rise of warmer fluid and the fall of cooler fluid. Forced convection: In forced convection, the fluid is forced to flow over a surface or in a tube by external means such as a pump, blower, or a fan. Newton’s law of cooling: It states that the rate of heat flow/ heat transfer from a solid surface of area A, at a temperature Tw to a fluid at a temperature Ts is Qconvection = hA (Ts-T). Convection heat transfer coefficient (h): The rate of heat transfer between a solid surface and a fluid per unit surface area and per unit temperature difference is called convection heat transfer coefficient (h). Qconvection = hA (Ts-T) Where, h = convection heat transfer coefficient, W/m2.0C Convection heat transfer coefficient strongly depends on the following fluid properties: (i) By decreasing dynamic viscosity, convection heat transfer coefficient can be increased. By increasing thermal conduction, K convection heat transfer coefficient can be (ii) increased. By increasing specific heat, Cp convection heat transfer coefficient can be increased. (iii) By increasing fluid velocity, V convection heat transfer coefficient can be increased. (iv) Convection heat transfer coefficient also depends on: (v) Surface geometry; Surface roughness; (vi) Type of fluid flow. (vii) Laminar and Turbulent flows: An essential first step in the treatment of any convection problem is to determine whether the boundary layer is laminar or turbulent. Surface friction and the convection transfer rates depend strongly on which of these conditions exists. Laminar flow: In the laminar flow, fluid motion is highly ordered and it is possible to identify streamlines along which particles move. Fluid motion along a streamline is characterized by velocity components in both the x and y directions. Turbulent flow: Fluid motion in the turbulent flow is highly irregular and is characterized by velocity fluctuations. These fluctuations enhance the transfer of momentum, energy and species, and hence increase surface friction as well as convection transfer rates. Fluid mixing resulting from the fluctuations makes turbulent boundary layer thickness larger and boundary layer profiles (velocity, temperature) flatter than in laminar flow. Transition flow: Transition flow occurs between laminar and turbulent flow. The transition from laminar to turbulent flow does not occur suddenly; rather, it occurs over some region in which the flow hesitates between laminar and turbulent flows before it becomes fully turbulent. Transition from laminar to turbulent depends on: (i) Surface geometry (ii) Surface roughness (iii) Free stream velocity (iv) Surface temperature (v) Type of fluid. The velocity profile is approximately parabolic in laminar flow and becomes flatter in the turbulent flow with a sharp drop near the surface. Effect of turbulence: (i) (ii) (iii) Intense mixing of the fluid. Enhance heat and momentum transfer between fluid particles. Increase conduction heat transfer rate. Types of flow: (i) (ii) Internal flow: The fluid is completely confined by the inner surfaces of the tube and there is limit on how much the boundary layer grows. External flow: The fluid has a free surface and thus the boundary layer over the surface is free to grow indefinitely. Velocity Boundary layer: The region of flow that develops from the leading edge of the plate in which the effects of viscosity are observed is called the boundary layer. Some arbitrary point is used to designate the y position where the boundary layer ends; this point is usually chosen as the y coordinate where the velocity becomes 99 percent of the free stream velocity. Fluid velocity at the surface of the plate is zero (because of no-slip condition), and gradually increases with distance from the plate. At a sufficiently large distance from the plate, the fluid velocity becomes equal to the ‘free stream velocity’ V. The region above the plate surface within which this change of velocity from zero to the free stream value occurs is called the boundary layer (velocity boundary layer) also called the hydrodynamic boundary layer. The thickness of this region is called the boundary layer thickness and is denoted by . The boundary layer thickness increases with the distance x from the leading edge of the plate, i.e. = (x). Initially, the boundary-layer development is laminar, but at some critical distance from the leading edge, depending on the flow field and fluid properties, small disturbances in the flow begin to become amplified, and a transition process takes place until the flow becomes turbulent. The turbulent-flow region may be pictured as a random churning action with chunks of fluid moving to and fro in all directions. The transition from laminar to turbulent flow occurs when u x u x 5 10 5 Where, u free stream velocity, m / sec x = distance from leading edge, m Kinematic viscosity, m2/sec Property variation with time in a turbulent boundary layer: The thermal boundary layer: A velocity boundary layer develops when there is fluid flow over a surface; a thermal boundary layer must develop if the fluid free stream and surface temperatures differ. Consider flow over an isothermal flat plate. At the leading edge the temperature profile is uniform, with T(y) = T∞. However, fluid particles that come into contact with the plate achieve thermal equilibrium at the plate’s surface temperature. In turn, these particles exchange energy with those in the adjoining fluid layer, and temperature gradients develop in the fluid. The region of the fluid in which these temperature gradients exist is the thermal boundary layer, and its thickness is defined as t . With increasing distance from the leading edge, the effects of heat transfer penetrate further into the free stream and the thermal boundary layer grows. Significance of the boundary layers: The velocity boundary layer is of extent x and is characterized by the presence of velocity gradient and shear stresses. The thermal boundary is of extent t (x) and is characterized by temperature gradients and heat transfer. The principle manifestations of the two boundary layers are, respectively, surface friction, convection heat transfer. The key boundary layer parameters are then the friction coefficient Cf and the heat transfer convection coefficient h, respectively. For flow over any surface, there will always exist a velocity boundary layer, and hence surface friction. However, a thermal boundary, and hence convection heat transfer, exists only if the surface and free stream temperatures differ. The mean velocity: The velocity various over the cross section and there is no well-defined free stream, it is necessary to work with a mean velocity u m when dealing with internal flows. This velocity is defined such that, when multiplied by the fluid density and the cross sectional area of the tube Ac, it provides the rate of mass flow through the tube. Hence m u m Ac . Flow in tubes: The fluid velocity in a tube changes from zero at the surface to a maximum velocity at the tube centre. A boundary layer develops at the entrance. Eventually the boundary layer fills the entire tube, and the flow is said to be fully developed. If the flow is laminar, a parabolic velocity profile is experienced. When the flow is turbulent, a blunter profile is observed. In a tube, the Reynolds number is again used as a criterion for laminar and turbulent flow. For 2300. The flow is usually observed to be turbulent and where d is the tube diameter. Again, a range of Reynolds numbers for transition may be observed, depending on the surface roughness and smoothness of the flow. The generally accepted range for transition is 2000 Re d 4000. Re d um d Energy balance for flow in a tube: The flow in a tube is completely enclosed, an energy balance may be applied to determine how the mean temperature Tm x various with position along the tube and how the total convection heat transfer qconv is related to the difference in temperature at the tube inlet and outlet. The rate of convection heat transfer to the fluid must equal the rate at which the fluid thermal energy increases plus the net rate at which work is done in moving the fluid through the control volume. Pressure gradient and Friction factor in fully developed flow: The pressure drop needed to sustain an internal flow because this parameter determines the pump, blower or fan power requirements. To determine the pressure drop, it is convenient to work with the Moody (or Darcy) friction factor, which is a dimensionless parameter defined dp D dx as f . Where f is the friction factor. um 2 2 Resistance to fluid flow, the pressure drop in the flow direction: m 50kPa 35 kPa The fanning friction factor: The fanning friction factor is called the friction coefficient, which is defined as C f Where s um 2 2 C f is the friction coefficient or drag coefficient, whose value in most cases is determined experimentally, and is the density of the fluid. The friction coefficient, in general, will vary with location along the surface. The mean or Bulk Fluid temperature: The mean or bulk temperature of the fluid at a given cross section is defined in terms of the thermal energy transported by the fluid as it moves past the cross section. The rate at which this transport occurs, E t , may be obtained by integrating the product of the mass flux u and the internal energy per unit mass Et u c p c T d Ac Ac c T p over the cross section. That is, Hence if a mean temperature is defined such that Et m c p Tm . Where C p is the specific heat of the fluid and m is the mass flow rate. The product m C p Tm AT any cross section along the tube represents the energy flow with the fluid at that cross section. In the absence of any work interactions (such as electric resistance heating) , The conservation of energy equation for the steady flow of a fluid in a tube can be expressed as Q m C p Te Ti .Where Ti and Te are the mean fluid temperature at the inlet and exit of the tube, respectively, and Q is the rate of heat transfer to or from the fluid. After all, the bulk temperature is the representative of the total energy of the flow at any particular location. The bulk temperature is used for overall energy balances on systems. The thermal conditions: The thermal conditions at the surface of a tube can usually be approximated with reasonable accuracy to be constant surface temperature (Ts = constant) or constant surface heat flux q s cons tan t . For example, the constant surface temperature condition is realized when a phase change process such as boiling or condensation occurs at the outer surface of a tube. The constant surface heat flux condition is realized when the tube is subjected to radiation or electric resistance heating uniformly from all directions. The convection heat flux at any location on the tube can be expressed as q hTs Tm , where h is the local heat transfer coefficient and T s and Tm are the surface and the mean fluid temperatures at that location. Therefore, where h = constant, the surface temperature Ts must change when qs constant, and the surface heat flux q s must change when T s = constant. Thus we may have either T s = constant or qs constant at the surface of a tube, but not both. Constant surface heat flux ( qs constant): In the case of qs constant, the rate of heat transfer can also be expressed as Q q s A m C p Te Ti Then the mean fluid temperature at the tube exit becomes Te Ti qs A . mCp Variation of the tube surface and the mean fluid temperature along the tube for the case of constant surface heat flux: Fully developed region Entry region Ts Te ∆T=Ts-Tm=qs/h Tm Ti Constant surface temperature (Ts = constant): In the case of Ts = constant, the rate of heat transfer is expressed as T Ti Te Ti Q hA Tln , where Tln e is the logarithmic mean temperature Ts Te Te Ln Ln Ts Ti Ti difference. Here, Ti Ts Ti and Te Ts Te are the temperature differences between the surface and the fluid at the inlet and the exit of the tube, respectively. Then the mean fluid temperature at the tube exit in this case can be determined from Te Ts Ts Ti e hA mCp . The variation of the mean fluid temperature along the tube for the case of constant surface Temperature: Ts=costant Ts ∆T=Ts-Tm Ti L Ti Ts=costant Hydrodynamic entry region, length and hydrodynamically developed region: The region from the tube inlet to the point at which the boundary layer merges at the centerline is called the hydrodynamic entry region, and the length of this region is called the hydrodynamic entry length. The region beyond the hydrodynamic entry region in which the velocity profile is fully developed and remains unchanged is called the hydrodynamically developed region. Hydrodynamic entry lengths Lh 0.05 Red Laminar flow Lh 10 d Turbulent flow Velocity profile in the fully developed region: Thermal entry region, length and thermally developed region: The region of flow over which the thermal boundary layer develops and reaches the tube centre is called the thermal entry region and the length of this region is called the thermal entry length. The region beyond the thermal entry region in which the dimensionless temperature profile remains unchanged is called thermally developed region. The region in which the flow is both hydrodynamically and thermally developed is called the fully developed flow. Thermal entry lengths Lt 0.05 re. Pr d laminar flow Lt 10 d Turbulent flow For Pr >> 1 and Lh < Lt Laminar flow. Axial Variation of the convection heat transfer coefficient for flow in a tube: Viscous-energy dissipation function: The energy equation in the rectangular co-ordinate system for a elemental control volume for steady, two dimensional (x, y) flow of an incompressible, constant-property fluid when consider for convection energy, conduction energy and viscous energy is determined as 2T 2T T T K 2 2 C p u v x x y y Where is the viscous-energy dissipation function and is defined as u 2 v 2 v u 2 2 x y x y The left hand side represents the net energy transfer due to mass transfer; on the right hand side the first term represent the conductive heat transfer, and the last term on the right hand side is the viscous-energy dissipation in the fluid due to internal fluid friction. Physical significance of the dimensionless parameters: The dimensionless parameters such as the Reynolds number, Nusselt number and Prandtl numbers are introduced and the physical significance of these dimensionless parameters in the interpretation of the conditions associated with fluid flow or heat transfer is discussed. The Reynolds number: The Reynolds number represents the ratio of the inertia to viscous force. This result implies that viscous forces are dominant for small Reynolds numbers and inertia forces are dominant for large Reynolds numbers. The Reynolds number is used as the criterion to determine whether the flow is laminar or turbulent. As the Reynolds number is increased, the inertia forces become dominant and small disturbances in the fluid may be amplified to cause the transition from laminar to turbulent. where: vs - mean fluid velocity, L - characteristic length (equal to diameter (2r) if a cross-section is circular), μ - (absolute) dynamic fluid viscosity, ν - kinematic fluid viscosity: ν = μ / ρ, ρ - fluid density. Nusselt number: The Nusselt number is a dimensionless number that measures the enhancement of heat transfer from a surface that occurs in a real situation, compared to the heat transferred if just conduction occurred. Typically it is used to measure the enhancement of heat transfer when convection takes place. where L = characteristic length, which is simply Volume of the body divided by the Area of the body (useful for more complex shapes) kf = thermal conductivity of the "fluid" h = convection heat transfer coefficient Thus the Nusselt number may be interpreted as the ratio of heat transfer by convection to conduction across the fluid layer of thickness L. Based on this interpretation, the value of the Nusselt number equal to unity implies that there is no convection-the heat transfer is by pure conduction. A large value of the Nusselt number implies enhanced heat transfer by convection. The Prandtl number: The Prandtl number is a dimensionless number approximating the ratio of momentum diffusivity and thermal diffusivity. The Prandtl number provides a measure of the relative effectiveness of momentum and energy transport by diffusion in the velocity and thermal boundary layers, respectively. where: ν is the kinematic viscosity, ν = μ / ρ. α is the thermal diffusivity, α = k / (ρ cp). In heat transfer problems, the Prandtl number controls the relative thickness of the momentum and thermal boundary layers. The Stanton number: The Stanton number is a dimensionless number which measures the ratio of heat transferred into a fluid to the thermal capacity of fluid. It is used to characterize heat transfer in forced convection flows. where; h = convection heat transfer coefficient ρ = density of the fluid cp = specific heat of the fluid V = velocity of the fluid It can also be represented in terms of the fluid's Nusselt, Reynolds, and Prandtl numbers: where; Nu is the Nusselt number Re is the Reynolds number Pr is the Prandtl number Heat transfer enhancement: Several options are available for enhancing heat transfer associated with internal flows. Enhancement may be achieved by increasing the convection coefficient and/or by increasing the convection surface area. For example, h may be increased by introducing surface roughness to enhance turbulence, as, for example, through machining or insertion of a coilspring wire. The wire insert provides a helical roughness element in contact with the tube inner surface. Alternatively, the convection coefficient may be increased by inducing swirl through insertion of a twisted tape. The insert consists of a thin strip that is periodically twisted through 3600. Introduction of a tangential velocity component increases the speed of the flow, particularly near the tube wall. The heat transfer area may be increased by attaching longitudinal fins to the inner surface, while both the convection coefficient and area may be increased by using spiral fins or ribs. In evaluating any heat transfer enhancement scheme, attention must also be given to the attendant increase in pressure drop and hence fan or pump power requirements.
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