Viscous Flow and Convective Heat Transfer (EGFD 7041) Fall 2018

Viscous Flow and Convective Heat Transfer (EGFD 7041) Fall 2018 Introduction & Review Dr. Peter J. Disimile UC-FEST Department of Aerospace Engineering Peter.disimile@uc.edu 1 2 1

1.Basic Modes of Heat Transfer Conduction transfer of internal energy from: - one body to another this takes place by the flow of free electrons (for metals) or by the transfer of kinetic energy of the molecules by direct contact with another body, - one location in a body to another transfer of energy caused by an interaction between atoms and molecules of a substance at different temperatures. Radiation transfer of thermal energy resulting from the transport of electromagnetic waves or photons. Convection energy transfer resulting from fluid motion. Although only one of the above modes of heat transfer may dominate in any engineering application, a combination of these modes are typically involved. 3 2. Convective Heat Transfer The transfer of thermal energy by bulk fluid motion (i.e., mixing), to (or from) a surface by the local fluid motion. Note: i) The process of energy transfer resulting from the gross movement of fluid is also known as advection. ii) However the final transfer of energy from one molecule or particle to another is ultimately due to thermal conduction. 4 2

Thermal Convection can be subdivided into: a) Natural or free convection motion generated as a result of density variations, which exists in a fluid due to it s temperature difference. Example: Heated Plate Plate is heated Gradient in temperature Gradient in density Gradient in pressure Transition Turbulent Symmetrical collision lines Heated Plate Laminar Side View Top View Multi dimensional flow over a heated plate 5 Heated Plate (Con t) Different thermal energy transfer mechanisms appear in the flow field above the heated plate. 6 3

b) Forced convection when fluid motion is a result of an external mechanical driving force such as a fan, blower, pump, etc. Example 1: Fans delivering warm air to the occupied space within a vehicle. Example 2: Fans blowing cooler air across heat sinks of a thermal generator. 7 3. Aerodynamic Heating When solid bodies undergo high-speed motion within a fluid, high temperatures within the boundary layer have been observed and are a result of aerodynamic heating. These high temperatures are generated as the flow is slowed down within the viscous region adjacent to the body. Such conditions are found in high speed flows related to the reentry of space vehicles, ballistic missiles and satellites. 8 4

In such situations the heat transfer coefficient (h) was determined to be proportional to two constants: - K1 for 3D flows and - K2 for 2D flows: 9 4. The Boundary Layer When considering real fluids, the fluid particles in contact with a solid surface have the same velocity as that surface. This is called the no-slip condition. Moving away from the surface the fluid velocity increases. This increase continues until the edge of the hydrodynamic boundary layer, as indicated by attainment of the freestream velocity, U. 10 5

The boundary layer thickness, is the distance between the edge of the hydrodynamic boundary layer and the solid surface. Unfortunately, the boundary edge is loosely defined as the location where the local velocity approximately reaches the freestream velocity, U. Common values for the boundary edge are around 99% U. 11 Likewise, a thermal boundary layer and its associated thickness, T can also be defined by the distance required for the local fluid temperature, T(y) to reach the freestream fluid temperature, T f. In the present example the surface temperature, T s > T f. 12 6

5. Newton s Law of Cooling Combining all the difficulties in conductive heat transfer and fluid mechanics, a single lumped parameter, modeled after the unit thermal conductance was introduced into the cooling law. This parameter is the convective heat transfer coefficient, h. h is a function of the fluid composition, thermophysical properties, surface geometry and fluid motion. q ha ( T T s s f where A s = surface area. h = convective heat transfer coefficient T s = surface temperature T f = fluid temperature ) (1.1) 13 A more general form of the Cooling Law dq h x da s ( T s T f ) (1.2) where dq = rate of heat transfer from a elemental surface area element, da s h x = the local heat transfer coefficient The mean and local heat transfer coefficient is related by; h 1 A s As h x da s (1.3) 14 7

Typical ranges of selected convective heat transfer coefficients are listed in the following table. System W / (m 2 0 C) Btu /(h ft 2 0 F) Natural Convection Air 5-30 0.9-5 Water 200-600 30-100 Forced Convection Air 100-500 2-100 Water 100 2x10 4 20 4x10 4 Oil 60 2x10 3 10-400 Phase Change Boiling Water 2x10 3 5x10 4 300-9x10 3 at 1 atm. Steam Condensation 5x10 3 1x10 5 900 2x10 4 h 15 6. Basic Concepts and Definitions: Solids - A substance which has a definite shape regardless of whether small to moderate shear forces are applied to its surface. (Molecules are closely positioned and have large intermolecular forces.) Fluids - Substances when at rest, cannot sustain a shear force (or tangential force). Liquids A state of matter in which molecules are relatively free to change their position with respect to each other but restricted by intermolecular (cohesive) forces, so as to maintain a relatively fixed volume. Gases A state of matter in which the molecules are practically unrestricted by intermolecular forces (molecules spaced relatively far apart). Hence, a gas has neither a definite shape nor volume. 16 8

Continuum When the properties of a fluid are considered to be continuously distributed throughout the region of interest, or when the dimensions of the problem are large w.r.t. the spacing between the molecules. Note: i) As the pressure is significantly reduced, the average distance between molecules becomes large compared to the dimensions of the object over which the fluid is flowing. Under these conditions the fluid is now considered a rarified gas & no longer fulfills the continuum assumption. ii) There are 2.7x10 16 molecules contained in a cubic millimeter of air at standard conditions (STD). iii) To determine if the continuum assumption is valid, compare the characteristic length, of the object under study to the molecular mean free path (), which is the average distance a molecule travels before it collides with another molecule. If >>, the continuum model is acceptable. For air at STD conditions, is 6x10-6 cm = 60 nm. 17 Body Forces are those forces which involve action from a distance, and are proportional to either the volume or mass of a body. Examples of body forces are those arising from: gravity magnetic fields electrodynamics Surface Forces - those forces which are exerted at the control surface by the material outside the control volume on the material inside the control volume. Examples of these surface forces are those arising from: normal stresses (or pressure), shear stresses (viscous or turbulent), surface tension (when interfaces between phases exist). Pressure - The force/unit area, exerted perpendicular to a surface. Density () - mass/unit volume; in general =f(t,p) 18 9

Table A.2 Properties of the U.S. Standard Atmosphere (EE & BG Units) Geometric Gravitational Kinematic Altitude Temp Pressure Density Density Acceleration Viscosity Viscosity z (ft) T ( R) p (psia) (lbm/ft3) (slug/ft3) g (ft/s2) (lb s/ft2) (ft2/s) -15000 572.2 24.626 1.162E-1 3.610E-3 32.220 4.031E-7 1.116E-4-10000 554.3 20.847 1.015E-1 3.155E-3 32.205 3.935E-7 1.247E-4-5000 536.5 17.554 8.831E-2 2.745E-3 32.189 3.835E-7 1.398E-4 0 518.7 14.696 7.647E-2 2.377E-3 32.174 3.736E-7 1.572E-4 5000 500.8 12.054 6.590E-2 2.048E-3 32.159 3.636E-7 1.776E-4 10000 483.0 10.108 5.648E-2 1.756E-3 32.143 3.534E-7 2.013E-4 20000 447.4 6.759 4.077E-2 1.267E-3 32.112 3.326E-7 2.623E-4 30000 411.8 4.373 2.866E-2 8.907E-4 32.082 3.107E-7 3.488E-4 40000 390.0 2.730 1.890E-2 5.873E-4 32.051 2.969E-7 5.057E-4 50000 390.0 1.692 1.171E-2 3.639E-4 32.020 2.969E-7 8.159E-4 60000 390.0 1.049 7.259E-3 2.256E-4 31.990 2.969E-7 1.316E-3 70000 392.2 0.651 4.479E-3 1.392E-4 31.959 2.983E-7 2.143E-3 80000 397.7 0.406 2.758E-3 8.571E-5 31.929 3.018E-7 3.521E-3 90000 403.1 0.255 1.710E-3 5.315E-5 31.898 3.052E-7 5.743E-3 100000 408.6 0.162 1.068E-3 3.318E-5 31.868 3.087E-7 9.302E-3 150000 479.1 0.020 1.112E-4 3.456E-6 31.716 3.512E-7 1.016E-1 200000 457.0 0.003 1.696E-5 5.270E-7 31.566 3.382E-7 6.416E-1 250000 351.8 0.000 2.263E-6 7.034E-8 31.42 2.721E-7 3.868E0 300000 332.9 0.000 1.488E-7 4.625E-9 31.27 2.593E-7 5.608E1 Note: i) ρ is a f(temperature, pressure) ii) The density of air at STD is ρ = 1.23 kg/m 3 (or 0.00238 slug/ft 3 ) at P = 101.33 kpa (14.696 psia) and T = 15C (or 59F). Note: Typical satellite in LEO is approx 160 to 2,000 kilometers 19 Incompressible fluid a fluid in which the is assumed constant Note: Air @ T=const only needs to change P from 101kPa to 119kPa to achieve the 17% change in density 20 10

Note: H 2 O density changes by approximately 4%, while air over the same T range (273 to 373 K) changes 37% Note: Hg density changes by < 2% as compared to 37% for air over the same T s 21 Viscosity a thermophysical property which represents the resistance to the sliding motion of one fluid layer over another. A fluid undergoes a continuous deformation (or strain) when subjected to a shear stress,. Relating the to the rate of deformation is accomplished using the absolute viscosity,, which is a property of the fluid. Deformation rate (i.e., strain rate) is the velocity gradient, and in 1D is du/dy. Therefore, the 1D shear stress relation. du dy (1.4) Absolute or Dynamic viscosity () lb sec / ft 2 Kinematic viscosity () ft 2 / sec or m 2 /sec 22 11

If the fluid is compressible and if significant changes in volume occur, an additional viscous stress coefficient will be required. This coefficient is called the second or bulk viscosity. If is independent of the velocity gradient (i.e., the rate of strain), that is if varies linearly with du the fluid is called Newtonian. dy Examples of Newtonian fluids are water, air, alcohol, gases and most petroleum products (where is practically independent of the velocity gradient). Note: The absolute viscosity () is in general a f(p,t); - although changes in with P are usually small, - changes due to T may be very large. - (see Figures. 3A & 4A and Table 1A.) 23 TABLE 1-A, Displays the effect of pressure on the absolute viscosities of water and a typical lubricating oil (Similar to SAE 30). The values in the table represent the absolute viscosity () at the a specified pressure divided by the () at 1 atmosphere. Substance Pressure in Atmospheres 100 300 500 750 1000 2000 Water at 30 0 C 1.0 1.01 1.02 1.04 1.05 1.13 Water at 10 0 C 1.0 0.98 0.97 0.96 0.95 0.965 Representative Lubricating Oil (~ SAE 30) at 55 0 C 1.45 2.50 4.7 9.4 19 ~ 150 Data adapted from International Critical Tables, McGraw-Hill Book Company, New York, 1926 (courtesy of the National Academy of Sciences, National Research Council, Washington, D.C.); H.A. Everett, High Pressure Viscosity as an Explanation of Apparent Oiliness, Soc. Aut. Eng., Trans., vol. 41, 5, p. 531, 1937; R. B. Dow, The Effect of Temperature and Pressure on the Viscosity of Lubricating Oils, Rheology Bulletin, Am. Inst. of Physics, 1937. 24 12

FIG. 3-A Absolute viscosity of various fluids. Sreferstothe density of the substance relative to water @ 60 deg F. Prepared from data in R.L. Daugherty and A. C. Ingersoll, Fluid Mechanics, McGraw-Hill Book Company, New York, 1954. FIG. 4-A Kinematic viscosity of various fluids. Shasthesame meaning as in Fig. 3-A. Prepared from data in R.L. Daugherty and A. C. Ingersoll, Fluid Mechanics, McGraw-Hill Book Company, New York, 1954. Note: 1- Air viscosity increased by 20% as T increased from 18 ºC to 100 ºC, however the viscosity of H 2 O decreases by almost a factor of 4 over the same T s. 2- In general gas s increase with T and the viscosity of liquids decrease with increasing T. 25 Newtonian vs. Non-Newtonian Fluids Air and water are considered to be Newtonian, however, not all fluids are considered Newtonian. Viscosity of non-newtonian fluids are a function of strain rate. In general, solutions containing long chain polymers, as well as blood, slurries and suspensions are usually considered Non-Newtonian. Non-Newtonian Fluid Examples 26 13

Newtonian vs. Non-Newtonian Fluids Non-Newton fluids can be divided into two major sub groups: Shear Thickening fluids that flow easy with low viscosity at low strain rates, but become more solid-like as the strain rate is increased. Shear Thinning fluids in which viscous effects decrease as the strain rate increases. c Shear thinning Newtonian (high μ) Newtonian (low μ) Shear thickening Strain rate (1/s) 27 Newtonian vs. Non-Newtonian Fluids Shear Thickening fluids flow easy with low viscosity at low strain rates but become more solid-like (very viscous) as the strain rate increases. Examples of shear thickening fluids are medium greases, sludge's, and corn starch/water mixtures. In the limit of the shear thickening fluids, is a group of fluids called dilatants ; they become more solidlike as the strain rate increases (i.e., quicksand). Fluids with even higher viscosity tend to behave as a solid and exhibit a plastic like behavior; they don t begin to flow until a finite stress has been reached (i.e., toothpaste and heavy greases). c Shear thinning Newtonian (high μ) Shear thickening Strain rate (1/s) 28 14

Newtonian vs. Non-Newtonian Fluids Shear Thinning fluids have viscous effects that decrease as the strain rate increases. Examples of shear thinning fluids are ketchup and most salad dressings (i.e., they come out of the bottle all at once). Also hair gel, where it s much harder to shake off your fingers (a low shear application), but has much less resistance when rubbed between the fingers (a high shear application). Some shear thinning fluids are referred to as pseudoplastic. These are usually solutions of large, polymeric molecules in a solvent with smaller molecules. c Shear thinning Newtonian (high μ) Newtonian (low μ) Shear thickening Strain rate (1/s) 29 Newtonian vs. Non-Newtonian Fluids A analytical approach to Non-Newtonian behavior uses the power law approximation, (not valid near the origin of the stress strain curves) n u n xy K 2K xy y where K is the flow consistency index and n is the flow behavior index, which are related to the specific substance. Parameters typically vary with T, p, and substance composition. n is the flow behavior index; n < 1 Pseudoplastic (Shear Thinning) n =1 Newtonian fluid (K = ) n >1 Dilatant (Shear Thickening) Note: The power law only provides a good description of fluid behavior over the range of shear rates to which coefficients were fitted. 30 15

VISCOSITY AS A FUNCTION OF T & P Since there are no single functional relations for (T,p) which describes a large class of fluids, non-dimensional form based on the critical point can provide +/- 20% accuracy. This is referred to as the principle of corresponding states: r c T r =T/T c ; p r =p/p c where r, T r, and p r are the reduced quantities, and c, T c, and p c are the critical quantities. General Trends 1. Viscosity of liquids as T 2. Viscosity of low-pressure gases (or dilute mixtures) as T 3. Viscosity always as P 4. Very poor accuracy near the critical point (Pc, Tc) - Usually Pc ~ 10 atm - Common to ignore P dependence in The liquid vapor critical point, is the location on the P T curve which many problems and only consider T represents conditions under which a liquid and vapor can coexist dependence 31 CORRELATIONS FOR AND k n T T T S and 0 0 T 0 0 T 0 T S 3 2 32 16

CORRELATIONS FOR AND k n 3 2 k k 0 and 0 k0 k T T 0 T T 0 T S T S 33 VISCOUS BEHAVIOR OF VARIOUS MATERIALS Some non-newtonian fluids may exhibit a time-dependent behavior, i.e., if the strain rate is held constant, the shear stress will vary. Non-Newtonian fluids Rheopectic Thixotropic Under Constant Strain rate, There are two groups of time dependent fluids: Thixotropic when shear stress decreases over time as the strain is applied; The longer these fluids are under a strain, the lower is the viscosity Examples: yogurt, paint, automatic transmissions fluids. Rheopectic when shear stress increases over the time as the strain is applied; The longer these fluids are strained, the higher is the viscosity Examples: Gypsum paste, and some lubricants which thicken or solidify when shaken. 34 17

Chapter 1C: Mathematical Concepts V 0 V V 0 2 0 If the curl of the velocity field is zero Flow is irrotational Velocity can also be written as the gradient of a scalar function, If the divergence of the velocity field is zero Flow is incompressible If true Laplace equation q 0 A 0 The curl of the gradient of a scalar function is zero The divergence of the curl of a vector is zero 35 Chapter 1C: Mathematical Concepts Gradient Theorem Vector equation involving a scalar function, q Limits of integration is set such that surface encloses the volume The unit normal vector, n points outward Divergence Theorem (Gauss ) The vector quantity is A Scalar equation results qd Ad S S qnˆds A nˆds Stokes Theorem Direction of n is given by right hand rule over the path length dl S A nˆds L A dl Due HW # 1 36 18