Heat Transfer External Convection

Introduction Examples of external flow include flow over a flat plate, and flow over curved surfaces such as cylinders, spheres, airfoils and turbines blades Primary objective in external flow is to determine convection coefficients for different flow geometries Solution methods include: Experimental or Empirical Method Theoretical Method Non-dimensionalized approach is generally followed, and the local and average convection coefficients are generally correlated by the following for of equations: Heat Transfer: Mass Transfer:

Experimental or Empirical Method The empirical method involves heat and mass transfer measurements and data correlation using appropriate dimensionless parameters For fixed Prandtl numbers for a given fluid, log-log plots of Re versus Nu generally are straight lines These lines collapse to a single line for all Pr when plotting log(Nu / Pr) vs log(Re). The constants C, m, and n for mass transfer (Sh) are the same as obtained for heat transfer for similar geometry C, m, n are constants that depend on geometry, …

The Flat Plate in Parallel Flow Laminar Flow Assuming Steady, incompressible, laminar flow with constant fluid properties and negligible viscous dissipation, and taking dP/dx = 0, the boundary layer equations become: Using the Blasius method, the continuity and momentum equations are solved to obtain u and v in terms of free stream functions ψ (x,y). The solution yields: Continuity: Energy: Momentum:

The flat plate in parallel flow Turbulent Laminar L U∞, T∞ x y xc Ts

And Ts = constant also Average Boundary Layer Parameters where Ts = constant i.e and and

Note: In all the above expressions, the effect of variable properties can be treated by evaluating all properties at the film temperature, Tf. Also, for liquid metals (Pr is very small, t>>) it is reasonable to assume uniform velocity U = U∞ throughout the entire thermal boundary layer. A single correlation that applies to all Pr values is given by Churchill and Ozoe for laminar flow over an isothermal plate: Where Pe is the Peclet Number Pex = Rex Pr. Ts = constant

(b) Turbulent Flow Friction Factor Velocity B.L. thickness Nusselt Number: Note: For laminar flow  varies with x½ (slower growth) Cf varies with x-½ (sharper decay) Ts = constant

(c) Mixed Boundary Layer Conditions: (laminar and turbulent flow over a plate) Using values of hlam and hturb from previous analysis we get: Turbulent Laminar L U∞, T∞ x y xc Ts or where

Similarly for the friction factor And Rex,c is the critical Reynolds number in the x-direction, For a typical value or Rex,c = 5x105, A = 871, and: Mixed B.L./turbulent B.L. Rex,c ≈ 5 x 105 , Ts = constant, 0.6 < Pr < 60, 0.6 < Sc < 3000, 5 x 105 < ReL < 108 Similarly for the friction factor i.e. For Rex,c = 5 x 105

When L>>xc, (ReL>>Rex,c), A>> 0.037 ReL4/5 Fluid properties in all correlations are estimated at Tf These equations may also be used when a turbulent boundary layer exists over the entire flat plate by using turbulent promoters. (d) Uniform Surface Heat Flux at the Plate Laminar Flow Turbulent Flow Ts = constant

Methodology for Convection Calculations Become immediately cognizant of the flow geometry. Does the problem involve flow over a flat plate, a sphere, or a cylinder? The specific form of the convection correlation depends, of course, on the geometry. Specify the appropriate reference temperature and evaluate the pertinent fluid properties at that temperature. For moderate boundary layer temperature differences, the film temperature (Ts+T)/2 may be used for this purpose. However, there are correlations that require property evaluation at T and include a property ratio to account for the non-constant property effect. For mass transfer, the pertinent fluid properties are those of species B. We will only consider dilute, binary mixtures (i.e., CA << CB) and therefore the properties of the mixture may be assumed to be the properties of species B. Schmidt number: Sc = B / DAB and Reynolds number: ReL = VL / B. Calculate the Reynolds number. Boundary layer conditions are strongly influenced by this parameter. If the geometry is a flat plate in parallel flow, determine whether the flow is laminar or turbulent. Decide whether a local of surface average coefficient is required. Recall that for constant surface temperature or vapor density, the local coefficient is used to determine the flux at a particular point on the surface, whereas the average coefficient determines the transfer rate for the entire surface. Select the appropriate correlation

The Cylinder in Cross Flow Boundary layer Flow considerations: Boundary layer formation and separation on a cylinder in a cross flow Forward Stagnation Point (U∞ = 0) Separation Point ∂u/∂y = 0 Favorable Pressure Gradient Adverse Pressure Gradient Velocity profile associated with separation on a cylinder in Cross flow Separation Point Flow Reversal

For a circular cylinder, Lc = D As boundary layer develops Nuθ ↓ as θ ↑ After transition or separation point Nuθ ↑ as θ ↑ For ReD ≤ 2 x 105 B.L. is laminar Separation occurs at θ ≈ 80° For ReD ≥ 2 x 105 B.L. is transitional Transition to turbulence at θ ≈ 80-100° Turbulent b.l. develops (θ ≈ 110-140°) Separation occurs at θ ≈ 140°

Drag Force / Coefficient, (FD/ CD) Acting on the cylinder Where FD = Drag force with two components→ Friction drag and Pressure Drag Af = Cylinder Frontal Area (projected area perpendicular to the free stream), = LD CD is a function of ReD as shown below:

Correlations for Nusselt Number, Nu accuracy within ± 20% Hilpert C, m are functions of ReD – values are given in table 7.2 All properties are estimated at Tf For gas flow over cylinders of non – circular cross section, values of C and m are given in table 7.3 Zhukauskas (newer data) – circular cylinder All properties are evaluated at T∞, except Prs at Ts Values of constants C and m are given in table 7.4 Churchill and Bernstein (newer data) Properties are estimated at Tf Covers all ranges of ReD and applicable for ReDPr > 2

The Sphere Flow over a sphere is much like in a circular cylinder Results for drag Coefficient are shown in Fig. 7.8 (text) For very small Re, the drag coefficient is given by Stoke’s Law Whitaker correlation All properties evaluated at T∞, except μs Results applicable to mass transfer problems by replacing NuD and Pr with ShD and Sc, respectively

Flow Across Banks of Tubes Applications include steam generation in a boiler and air cooling in the coil of an air conditioner One fluid moves over the tubes, and the second fluid passes through the tubes Tube rows either staggered or aligned in the direction of fluid velocity V Configuration characterized by transverse pitch ST and longitudinal Pitch SL Generally we are interested in average heat transfer coefficient for entire tube budle Aligned Staggered

Heat Transfer Correlations Grimson For airflow (Pr=0.7) across bundles of 10 or more rows (NL≥ 10): Modification for other fluids across tube bundles: Properties at Tf Number of rows NL≥ 10 2000 < ReD,max < 40,000 Pr ≥ 0.7 C1 and m depend on geometry – values are listed in table 7.5 For NL< 10 use correction factor C2 from table 7.6 aligned staggered

Zhukauskas (more recent correlation) Properties at Tm (except Prs) 0.7 < Pr < 500 1000 < ReD,max < 2 x 106 NL ≥ 20 C, m values are given in table 7.7 For NL< 20 use correction factor C2 from table 7.8

Normally, Newton’s Law of Cooing gives Heat Transfer Rate Normally, Newton’s Law of Cooing gives The fluid experiences a change in temperature as it moves through the tube bank. As a result, the temperature difference ∆T decreases. The appropriate form for ∆T is the logarithmic mean temperature difference, given by: The heat transfer rate is therefore: The outlet temperature of the fluid To could be estimated from: heat transfer rate per unit length of tubes Where N = total number of tubes NT= number of tubes in transverse direction

Temperature and Heat Flux Variation: Typically, h for the aligned tubes increases with increasing the row number until ≈ 5th row. (Moderate values of SL) For ST / SL < 0.7, much of the flow through downstream rows is shielded by flow through upstream row, and h decreases (undesirable). In this case, the preferred flow path is in lanes between the tubes and much of the tube surface is not exposed to the main flow Heat transfer enhancement, in general, is favored by more tortuous flow, flow of a staggered arrangement, particularly at small Reynolds numbers (ReD < 100). Aligned Staggered

Pressure Drop Across a Tube Bank Power (P=Q ∆P) to move the fluid across is often a major expense and is proportional to the pressure drop, ∆P. f = friction factor and  = correction factor fig 7.13 for in line tube arrangement fig 7.14 for staggered tube arrangement

Summary of correlations for external flow

Problem 7.10 Problem 7.42

Problem 7.65 Problem 7.133