UNIT II CONVECTION HEAT TRANSFER


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1 UNIT II CONVECTION HEAT TRANSFER Convection is the mode of heat transfer between a surface and a fluid moving over it. The energy transfer in convection is predominately due to the bulk motion of the fluid particles; through the molecular conduction within the fluid itself also contributes to some extent. If this motion is mainly due to the density variations associated with temperature gradients within the fluid, the mode of heat transfer is said to be due to free or natural convection. On the other hand if this fluid motion is principally produced by some superimposed velocity field like fan or blower, the energy transport is said to be due to forced convection. Common classifications: A. Based on geometry: External flow / Internal flow B. Based on driving mechanism Natural convection / forced convection / mixed convection C. Based on number of phases Single phase / multiple phase D. Based on nature of flow Laminar / turbulent Convection Boundary Layers: Velocity Boundary Layer: Consider the flow of fluid over a flat plate as shown in the figure. The fluid approaches the plate in x direction with uniform velocity u. The fluid particles in the fluid layer adjacent to the surface get zero velocity. This motionless layer acts to retard the motion of particles in the adjoining fluid layer as a result of friction between the particles of these two adjoining fluid layers at two different velocities. This fluid layer then acts to retard the motion of particles of next fluid layer and so on, until a distance y =d from the surface reaches, where these effects become negligible and the fluid velocity u reaches the free stream velocity u. as a result of frictional effects between the fluid layers, the local fluid velocity u will vary from x =0, y = 0 to y =d. The region of the flow over the surface bounded by d in which the effects of viscous shearing forces caused by fluid viscosity are observed, is called velocity boundary layer or hydro dynamic boundary layer. The thickness of boundary layer d is generally defined as a distance from the surface at which local velocity u = 0.99 of free stream velocity u. The retardation of fluid motion in the boundary layer is due to the shear stresses acting in opposite direction with increasing the distance y from the surface shear stress decreases, the local velocity u increases until approaches u. With increasing the distance from the leading edge, the effect of viscosity penetrates further into the free stream and boundary layer thickness grows.
2 Thermal boundary Layer: If the fluid flowing on a surface has a different temperature than the surface, the thermal boundary layer developed is similar to the velocity boundary layer. Consider a fluid at a temperature T flows over a surface at a constant temperature T s. The fluid particles in adjacent layer to the plate get the same temperature that of surface. The particles exchange heat energy with particles in adjoining fluid layers and so on. As a result, the temperature gradients are developed in the fluid layers and a temperature profile is developed in the fluid flow, which ranges from T s at the surface to fluid temperature T sufficiently far from the surface in y direction. Development of velocity boundary layer on a flat plate: It is most essential to distinguish between laminar and turbulent boundary layers. Initially, the boundary layer development is laminar as shown in figure for the flow over a flat plate. Depending upon the flow field and fluid properties, at some critical distance from the leading edge small disturbances in the flow begin to get amplified, a transition process takes place and the flow becomes turbulent. In laminar boundary layer, the fluid motion is highly ordered whereas the motion in the turbulent boundary layer is highly irregular with the fluid moving to and from in all directions. Due to fluid mixing resulting from these macroscopic motions, the turbulent boundary layer is thicker and the velocity profile in turbulent boundary layer is flatter than that in laminar flow. The critical distance xc beyond which the flow cannot retain its laminar character is usually specified in term of critical Reynolds number Re. Depending upon
3 surface and turbulence level of free stream the critical Reynolds number varies between 10 5 and 3 X In the turbulent boundary layer, as seen three distinct regimes exist. A laminar sublayer, existing next to the wall, has a nearly linear velocity profile. The convective transport in this layer is mainly molecular. In the buffer layer adjacent to the sublayer, the turbulent mixing and diffusion effects are comparable. Then there is the turbulent core with large scale turbulence. Dimensional analysis Dimensional analysis is a mathematical method which makes use of the study of the dimensions for solving several engineering problems. This method can be applied to all types of fluid resistances, heat flow problems in fluid mechanics and thermodynamics. Buckingham Π theorem. Buckingham Π theorem states as follows: "If there are n variables in a dimensionally homogeneous equation and if these contain m fundamental dimensions, then the variables are arranged into (n  m) dimensionless terms. These dimensionless terms are called Π terms. Advantages of dimensional analysis 1. It expresses the functional relationship between the variables in dimensional terms. 2. It enables getting up a theoretical solution in a simplified dimensionless form. 3. The results of one series of tests can be applied to a large number of other similar problems with the help of dimensional analysis. Limitations of dimensiona1 analysis 1. The complete information is not provided by dimensional analysis. It only indicates that there is some relationship between the parameters. 2. No information is given about the internal mechanism of physical phenomenon. 3. Dimensional analysis does not give any clue regarding the selection of variables. DIMENSIONLESS NUMBERS AND THEIR SIGNIFICANCE 1.Reynolds number (Re). It is defined as the ratio of inertia force to viscous force.
4 Significance of Reynolds number: Whether the flow is viscous dominance or inertial dominance is identified by Reynolds number. 2.Prandtl number (Pr). It is the ratio of the momentum diffusivity to the thermal diffusivity. Significance: It provides a measure of the relative effectiveness of the momentum and energy transport by diffusion. The Prandtl numbers of gases are about 1, which indicates that both momentum and heat dissipate through the fluid at about the same rate. Heat diffuses very quickly in liquid metals (Pr <<1) and very slowly in oils (Pr >> 1) relative to momentum 3.Nusselt Number (Nu). It is defined as the ratio of the heat flow by convection process under a unit temperature gradient to the heat flow rate by conduction under a unit temperature gradient through a stationary thickness (L) of meter. Significance: It is a measure of Forced convection. The larger the Nusselt number, the more effective the convection. Nu = 1 for a fluid layer represents heat transfer across the layer by pure conduction 4.Grashof number (Gr). It is defined as the ratio of product of inertia force and buoyancy force to the square of viscous force. Significance: It is a measure of Free convection 5.Stanton number (St). It is the ratio of Nusselt number to the product of Reynolds number and Prandtl number. Forced Convection Heat Transfer Convection is the mechanism of heat transfer through a fluid in the presence of bulk fluid motion. Convection is classified as natural (or free) and forced convection depending on how the fluid motion is initiated. In natural convection, any fluid motion is caused by natural means such as the buoyancy effect,
5 i.e. the rise of warmer fluid and fall the cooler fluid. Whereas in forced convection, the fluid is forced to flow over a surface or in a tube by external means such as a pump or fan. Mechanism of Forced Convection Convection heat transfer is complicated since it involves fluid motion as well as heat conduction. The fluid motion enhances heat transfer (the higher the velocity the higher the heat transfer rate). The rate of convection heat transfer is expressed by Newton s law of cooling: The convective heat transfer coefficient h strongly depends on the fluid properties and roughness of the solid surface, and the type of the fluid flow (laminar or turbulent). It is assumed that the velocity of the fluid is zero at the wall, this assumption is called no slip condition. As a result, the heat transfer from the solid surface to the fluid layer adjacent to the surface is by pure conduction, since the fluid is motionless. Thus, The convection heat transfer coefficient, in general, varies along the flow direction. The mean or average convection heat transfer coefficient for a surface is determined by (properly) averaging the local heat transfer coefficient over the entire surface. Flow Over Flat Plate The friction and heat transfer coefficient for a flat plate can be determined by solving the conservation of mass, momentum, and energy equations (either approximately or numerically). They can also be measured experimentally. It is found that the Nusselt number can be expressed as:
6 where C, m, and n are constants and L is the length of the flat plate. The properties of the fluid are usually evaluated at the film temperature defined as: Laminar Flow The local friction coefficient and the Nusselt number at the location x for laminar flow over a flat plate are where x is the distant from the leading edge of the plate and Rex = ρv x / μ. The averaged friction coefficient and the Nusselt number over the entire isothermal plate for laminar regime are: Taking the critical Reynolds number to be 5 x10 5, the length of the plate x cr over which the flow is laminar can be determined from Turbulent Flow The local friction coefficient and the Nusselt number at location x for turbulent flow over a flat isothermal plate are: The averaged friction coefficient and Nusselt number over the isothermal plate in turbulent region are:
7 Combined Laminar and Turbulent Flow If the plate is sufficiently long for the flow to become turbulent (and not long enough to disregard the laminar flow region), we should use the average values for friction coefficient and the Nusselt number. Example 1 Engine oil at 60 C flows over a 5 m long flat plate whose temperature is 20 C with a velocity of 2 m/s. Determine the total drag force and the rate of heat transfer per unit width of the entire plate. We assume the critical Reynolds number is 5x10 5. The properties of the oil at the film temperature are: The Re number for the plate is: Re L = V L / ν = 4.13x10 4
8 which is less than the critical Re. Thus we have laminar flow. The friction coefficient and the drag force can be found from: The Nusselt number is determined from: Flow across Cylinders and Spheres The characteristic length for a circular tube or sphere is the external diameter, D, and the Reynolds number is defined: The critical Re for the flow across spheres or tubes is 2x10 5. The approaching fluid to the cylinder (a sphere) will branch out and encircle the body, forming a boundary layer.
9 At low Re (Re < 4) numbers the fluid completely wraps around the body. At higher Re numbers, the fluid is too fast to remain attached to the surface as it approaches the top of the cylinder. Thus, the boundary layer detaches from the surface, forming a wake behind the body. This point is called the separation point. To reduce the drag coefficient, streamlined bodies are more suitable, e.g. airplanes are built to resemble birds and submarine to resemble fish, Fig. In flow past cylinder or spheres, flow separation occurs around 80 for laminar flow and 140 for turbulent flow. where frontal area of a cylinder is AN = L D, and for a sphere is AN = πd 2 / 4. The drag force acting on a body is caused by two effects: the friction drag (due to the shear stress at the surface) and the pressure drag which is due to pressure differential between the front and rear side of the body.as a result of transition to turbulent flow, which moves the separation point further to the rear of the body, a large reduction in the drag coefficient occurs. As a result, the surface of golf balls is intentionally roughened to induce turbulent at a lower Re number, see Fig.
10 The average heat transfer coefficient for cross flow over a cylinder can be found from the correlation presented by Churchill and Bernstein: For flow over a sphere, Whitaker recommended the following: Example 2
11 The decorative plastic film on a copper sphere of 10 mm diameter is cured in an oven at 75 C. Upon removal from the oven, the sphere is subjected to an air stream at 1 atm and 23 C having a velocity of 10 m/s, estimate how long it will take to cool the sphere to 35 C. Assumptions: 1. Negligible thermal resistance and capacitance for the plastic layer. 2. Spatially isothermal sphere. 3. Negligible Radiation. The time required to complete the cooling process may be obtained from the results for a lumped capacitance. Whitaker relationship can be used to find h for the flow over sphere: The required time for cooling is then
12 INTERNAL FORCED CONVECTION: The difference between pipe, duct and tubes are given below, Pipe  circular cross section. Duct  noncircular cross section. Tubes  smalldiameter pipes. The fluid velocity changes from zero at the surface (noslip) to a maximum at the pipe center. It is convenient to work with an average velocity, which remains constant in incompressible flow when the crosssectional area is constant. Laminar and Turbulent Flow in Tubes For flow in a circular tube, the Reynolds number is defined as For flow through noncircular tubes D is replaced by the hydraulic diameter D h. laminar flow: Re<2300 Transitional flow: 2300 Re 10,000 fully turbulent flow : Re>10,000. The Entrance Region A fluid entering a circular pipe at a uniform velocity. The noslip condition  the flow in a pipe is divided into two regions: Irrotational flow, Boundary layer The thickness of this boundary layer increases in the flow direction until it reaches the pipe center.
13 Hydrodynamic entrance region  the region from the pipe inlet to the point at which the boundary layer merges at the centerline. Hydrodynamically fully developed region The region beyond the entrance region in which the velocity profile is fully developed and remains unchanged. The velocity profile in the fully developed region is parabolic in laminar flow, and somewhat flatter or fuller in turbulent flow. Laminar Flow in Noncircular Tubes
14 Developing Laminar Flow in the Entrance Region For a circular tube of length L subjected to constant surface temperature, the average Nusselt number for the thermal entrance region (hydrodynamically developed flow) For flow between isothermal parallel plates
15 Flow Across Tube Banks Flow across tube banks depends on Tube diameter Longitudinal pitch S L. Transverse pitch S T. Number of tube rows deep N L. Number of tube columns wide N T. Tube arrangement. Flow rate. Fluid. Tube arrangements In line Staggered Geometric Considerations L = N L S L A 1 = N T S T Z V = L A 1 = N S L S T Z N = N L N T A s = Surface area = D N Z For staggered arrangement S D 2 = S L 2 + (S T /2) 2 A T = Transverse flow area = (S T D) N T Z A D = Diagonal Flow area = 2(S D D) N T Z
16 Surface area density = A s /V = D/( S L S T ) Minimum Flow area: Where: A min = C A N T D Z C A : Area coefficient = (S T /D  1) in line arrangement or staggered with S D S T D 2 = 2(S D /D1) staggered with S D S T D 2 Correlations for the heat transfer coefficient, h 0.36 Nu Pr for N L > 20 D C Re m D Nu D 0.36 C Re Pr F for N L < 20 m D c Natural Convection In natural convection, the fluid motion occurs by natural means such as buoyancy. Since the fluid velocity associated with natural convection is relatively low, the heat transfer coefficient encountered in natural convection is also low. Mechanisms of Natural Convection Consider a hot object exposed to cold air. The temperature of the outside of the object will drop (as a result of heat transfer with cold air), and the temperature of adjacent air to the object will rise. Consequently, the object is surrounded with a thin layer of warmer air and heat will be transferred from this layer to the outer layers of air. The temperature of the air adjacent to the hot object is higher, thus its density is lower. As a result, the heated air rises. This movement is called the natural convection current. Note that in the absence of this movement, heat transfer would be by conduction only and its rate would be much lower. In a gravitational field, there is a net force that pushes a light fluid placed in a heavier fluid upwards. This force is called the buoyancy force.
17 The magnitude of the buoyancy force is the weight of the fluid displaced by the body. Fbuoyancy = ρ fluid g V body where Vbody is the volume of the portion of the body immersed in the fluid. The net force is: Fnet = W Fbuoyancy Fnet = (ρ body ρ fluid ) g V body Note that the net force is proportional to the difference in the densities of the fluid and the body. This is known as Archimedes principle. We all encounter the feeling of weight loss in water which is caused by the buoyancy force. Other examples are hot balloon rising, and the chimney effect. Note that the buoyancy force needs the gravity field, thus in space (where no gravity exists) the buoyancy effects does not exist. Density is a function of temperature, the variation of density of a fluid with temperature at constant pressure can be expressed in terms of the volume expansion coefficient β, defined as: It can be shown that for an ideal gas where T is the absolute temperature. Note that the parameter βδt represents the fraction of volume change of a fluid that corresponds to a temperature change ΔT at constant pressure. Since the buoyancy force is proportional to the density difference, the larger the temperature difference between the fluid and the body, the larger the buoyancy force will be. Whenever two bodies in contact move relative to each other, a friction force develops at the contact surface in the direction opposite to that of the motion. Under steady conditions, the air flow rate driven by buoyancy is established by balancing the buoyancy force with the frictional force. Natural convection heat transfer on a surface depends on geometry, orientation, variation of temperature on the surface, and thermophysical properties of the fluid.
18 Grashof Number Grashof number is a dimensionless group. It represents the ratio of the buoyancy force to the viscous force acting on the fluid: It is also expressed as where g = gravitational acceleration, m/s 2 β = coefficient of volume expansion, 1/K δ = characteristic length of the geometry, m ν = kinematics viscosity of the fluid, m 2 /s The role played by Reynolds number in forced convection is played by the Grashof number in natural convection. The critical Grashof number is observed to be about 10 9 for vertical plates. Thus, the flow regime on a vertical plate becomes turbulent at Grashof number greater than The heat transfer rate in natural convection is expressed by Newton s law of cooling as: Q conv = h A (Ts T ) Natural Convection over Surfaces Natural convection on a surface depends on the geometry of the surface as well as its orientation. It also depends on the variation of temperature on the surface and the thermophysical properties of the fluid.
19 Note that the velocity at the edge of the boundary layer becomes zero. It is expected since the fluid beyond the boundary layer is stationary. The shape of the velocity and temperature profiles, in the cold plate case, remains the same but their direction is reversed. Natural Convection Correlations The complexities of the fluid flow make it very difficult to obtain simple analytical relations for natural convection. Thus, most of the relationships in natural convection are based on experimental correlations. The Rayleigh number is defined as the product of the Grashof and Prandtl numbers: The Nusselt number in natural convection is in the following form: where the constants C and n depend on the geometry of the surface and the flow. These relationships are for isothermal surfaces, but could be used approximately for the case of nonisothermal surfaces by assuming surface temperature to be constant at some average value. Isothermal Vertical Plate For a vertical plate, the characteristic length is L. Note that for ideal gases, β=1 / T Isothermal Horizontal Plate The characteristics length is A/p where the surface area is A, and perimeter is p. a) Upper surface of a hot plate b) Lower surface of a hot plate Example 1: isothermal vertical plate A large vertical plate 4 m high is maintained at 60 C and exposed to atmospheric air at 10 C. Calculate the heat transfer if the plate is 10 m wide.
20 Solution: We first determine the film temperature as Tf = (Ts + T ) / 2 = 35 C = 308 K The properties are: β = 1 / 308 = 3.25x103, k = (W/mK), ν = 16.5x106, Pr = 0.7 The Rayleigh number can be found: The Nusselt number can be found from: The heat transfer coefficient is Vertical Cylinders Use same correlations for vertical flat plate if: D ~ 35 1/ L Gr L Long Horizontal Cylinder hd n NuD CRa D k 4 Constants for general Nusselt number Equation
21
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