# fully developed laminar flow

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Chapter 19 FORCED CONVECTIONCopyright © 2012 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.Fundamentals of Thermal-Fluid Sciences 4th Edition in SI Units Yunus A. Çengel, John M. Cimbala, Robert H. Turner McGraw-Hill, 2012Lecture slides by Mehmet Kanoğlu

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*ObjectivesUnderstand the physical mechanism of convection and its classification Visualize the development of velocity and thermal boundary layers during flow over surfaces Gain a working knowledge of the dimensionless Reynolds, Prandtl, and Nusselt numbers Develop an understanding of the mechanism of heat transfer in turbulent flow. Evaluate the heat transfer associated with flow over a flat plate for both laminar and turbulent flow, and flow over cylinders and spheres. Have a visual understanding of different flow regions in internal flow, and calculate hydrodynamic and thermal entry lengths. Analyze heating and cooling of a fluid flowing in a tube under constant surface temperature and constant surface heat flux conditions, and work with the logarithmic mean temperature difference. Determine the Nusselt number in fully developed turbulent flow using empirical relations, and calculate the heat transfer rate.

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*19-1 PHYSICAL MECHANISM OF CONVECTIONConduction and convection both require the presence of a material medium but convection requires fluid motion. Convection involves fluid motion as well as heat conduction. Heat transfer through a solid is always by conduction. Heat transfer through a fluid is by convection in the presence of bulk fluid motion and by conduction in the absence of it. Therefore, conduction in a fluid can be viewed as the limiting case of convection, corresponding to the case of quiescent fluid.

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*The fluid motion enhances heat transfer, since it brings warmer and cooler chunks of fluid into contact, initiating higher rates of conduction at a greater number of sites in a fluid. The rate of heat transfer through a fluid is much higher by convection than it is by conduction. In fact, the higher the fluid velocity, the higher the rate of heat transfer.Heat transfer through a fluid sandwiched between two parallel plates.

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*Convection heat transfer strongly depends on the fluid properties dynamic viscosity, thermal conductivity, density, and specific heat, as well as the fluid velocity. It also depends on the geometry and the roughness of the solid surface, in addition to the type of fluid flow (such as being streamlined or turbulent).Convection heat transfer coefficient, h: The rate of heat transfer between a solid surface and a fluid per unit surface area per unit temperature difference.Newton’s law of cooling

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*The development of a velocity profile due to the no-slip condition as a fluid flows over a blunt nose.A fluid flowing over a stationary surface comes to a complete stop at the surface because of the no-slip condition.No-slip condition: A fluid in direct contact with a solid “sticks” to the surface due to viscous effects, and there is no slip. Boundary layer: The flow region adjacent to the wall in which the viscous effects (and thus the velocity gradients) are significant. The fluid property responsible for the no-slip condition and the development of the boundary layer is viscosity.

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*An implication of the no-slip condition is that heat transfer from the solid surface to the fluid layer adjacent to the surface is by pure conduction, since the fluid layer is motionless, and can be expressed asThe determination of the convection heat transfer coefficient when the temperature distribution within the fluid is knownThe convection heat transfer coefficient, in general, varies along the flow (or x-) direction. The average or mean convection heat transfer coefficient for a surface in such cases is determined by properly averaging the local convection heat transfer coefficients over the entire surface area As or length L as

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*Nusselt NumberHeat transfer through a fluid layer of thickness L and temperature difference T.In convection studies, it is common practice to nondimensionalize the governing equations and combine the variables, which group together into dimensionless numbers in order to reduce the number of total variables. Nusselt number: Dimensionless convection heat transfer coefficientLc characteristic lengthThe Nusselt number represents the enhancement of heat transfer through a fluid layer as a result of convection relative to conduction across the same fluid layer. The larger the Nusselt number, the more effective the convection. A Nusselt number of Nu = 1 for a fluid layer represents heat transfer across the layer by pure conduction.

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*We turn on the fan on hot summer days to help our body cool more effectively. The higher the fan speed, the better we feel. We stir our soup and blow on a hot slice of pizza to make them cool faster. The air on windy winter days feels much colder than it actually is. The simplest solution to heating problems in electronics packaging is to use a large enough fan.Convection in daily life

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**19-2 THERMAL BOUNDARY LAYERThermal boundary layer on a flat plate (the fluid is hotter than the plate surface).A thermal boundary layer develops when a fluid at a specified temperature flows over a surface that is at a different temperature. Thermal boundary layer: The flow region over the surface in which the temperature variation in the direction normal to the surface is significant. The thickness of the thermal boundary layer t at any location along the surface is defined as the distance from the surface at which the temperature difference T − Ts equals 0.99(T− Ts).The thickness of the thermal boundary layer increases in the flow direction, since the effects of heat transfer are felt at greater distances from the surface further down stream. The shape of the temperature profile in the thermal boundary layer dictates the convection heat transfer between a solid surface and the fluid flowing over it.

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**Prandtl NumberThe relative thickness of the velocity and the thermal boundary layers is best described by the dimensionless parameter Prandtl numberThe 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. Consequently the thermal boundary layer is much thicker for liquid metals and much thinner for oils relative to the velocity boundary layer.

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*19-3 PARALLEL FLOW OVER FLAT PLATESThe transition from laminar to turbulent flow depends on the surface geometry, surface roughness, upstream velocity, surface temperature, and the type of fluid, among other things, and is best characterized by the Reynolds number. The Reynolds number at a distance x from the leading edge of a flat plate is expressed asA generally accepted value for the Critical Reynold numberThe actual value of the engineering critical Reynolds number for a flat plate may vary somewhat from 105 to 3  106, depending on the surface roughness, the turbulence level, and the variation of pressure along the surface.

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*The variation of the local friction and heat transfer coefficients for flow over a flat plate.The local Nusselt number at a location x for laminar flow over a flat plate may be obtained by solving the differential energy equation to beThe local friction and heat transfer coefficients are higher in turbulent flow than they are in laminar flow. Also, hx reaches its highest values when the flow becomes fully turbulent, and then decreases by a factor of x−0.2 in the flow direction.These relations are for isothermal and smooth surfaces

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*Laminar + turbulentGraphical representation of the average heat transfer coefficient for a flat plate with combined laminar and turbulent flow.Nusselt numbers for average heat transfer coefficients

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*Flat Plate with Unheated Starting LengthFlow over a flat plate with an unheated starting length.Local Nusselt numbersAverage heat transfer coefficients

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*Uniform Heat FluxFor a flat plate subjected to uniform heat fluxThese relations give values that are 36 percent higher for laminar flow and 4 percent higher for turbulent flow relative to the isothermal plate case.When heat flux is prescribed, the rate of heat transfer to or from the plate and the surface temperature at a distance x are determined from

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*Flows across cylinders and spheres, in general, involve flow separation, which is difficult to handle analytically. Flow across cylinders and spheres has been studied experimentally by numerous investigators, and several empirical correlations have been developed for the heat transfer coefficient.Variation of the local heat transfer coefficient along the circumference of a circular cylinder in cross flow of air.19-4 FLOW OVER CYLINDERS AND SPHERES

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*The relations for cylinders above are for single cylinders or cylinders oriented such that the flow over them is not affected by the presence of others. They are applicable to smooth surfaces.

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**19-5 GENERAL CONSIDERATIONS FOR PIPE FLOWLiquid or gas flow through pipes or ducts is commonly used in heating and cooling applications and fluid distribution networks. The fluid in such applications is usually forced to flow by a fan or pump through a flow section. Although the theory of fluid flow is reasonably well understood, theoretical solutions are obtained only for a few simple cases such as fully developed laminar flow in a circular pipe. Therefore, we must rely on experimental results and empirical relations for most fluid flow problems rather than closed-form analytical solutions.Circular pipes can withstand large pressure differences between the inside and the outside without undergoing any significant distortion, but noncircular pipes cannot.For a fixed surface area, the circular tube gives the most heat transfer for the least pressure drop.

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**In fluid flow, it is convenient to work with an average or mean temperature Tm, which remains constant at a cross section. The mean temperature Tm changes in the flow direction whenever the fluid is heated or cooled.When designing piping networks and determining pumping power, a conservative approach is taken and flows with Re > 4000 are assumed to be turbulent.Re < 2300 laminar, Re > 10,000 turbulent, and transitional in between. Bulk mean fluid temperature

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*Thermal Entrance Region

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*The fluid properties in internal flow are usually evaluated at the bulk mean fluid temperature, which is the arithmetic average of the mean temperatures at the inlet and the exit: Tb = (Tm, i + Tm, e)/2The development of the thermal boundary layer in a tube.Thermal entrance region: The region of flow over which the thermal boundary layer develops and reaches the tube center. Thermal entry length: The length of this region. Thermally developing flow: Flow in the thermal entrance region. This is the region where the temperature profile develops. Thermally fully developed region: The region beyond the thermal entrance region in which the dimensionless temperature profile remains unchanged. Fully developed flow: The region in which the flow is both hydrodynamically and thermally developed.

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*In the thermally fully developed region of a tube, the local convection coefficient is constant (does not vary with x). Therefore, both the friction (which is related to wall shear stress) and convection coefficients remain constant in the fully developed region of a tube. The pressure drop and heat flux are higher in the entrance regions of a tube, and the effect of the entrance region is always to increase the average friction factor and heat transfer coefficient for the entire tube.Variation of the friction factor and the convection heat transfer coefficient in the flow direction for flow in a tube (Pr>1).Hydrodynamically fully developed:Thermally fully developed:Surface heat flux

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Last Updated: 8th March 2018