Chapter 7: External Flow

1. External flow • Flow boundary layers develop freely, without constraints imposed by adjacent surfaces • There will always exist a region of the flow outside the boundary layer in which velocity and temperature gradients are negligible Chapter 7: External Flow Restrictions: Low speed, forced convection, with no change of phase Objective: Determine convection coefficients for different flow geometries

2. Some Important Dimensionless Numbers for Heat Transfer

3. Boundary Layer Relations

4. Two Approaches for Heat Transfer From Chapter 6: Averaged from x = 0, where the boundary layer begins to develop, to the location of interest, x Two approaches: Experimental, or empirical approach Theoretical approach Theoretical approach: Solving the boundary layer equations for a particular geometry; limited to laminar flows and simple geometries

5. Experimental or Semi-empirical Approach Performing heat transfer measurements under controlled laboratory conditions and correlating the data in terms of appropriate dimensionless parameters The experimental or empirical approach will be emphasized in this chapter. The convection heat transfer coefficient may be correlated by equation of the form: It indicates that the averaged Nusselt number is a function of Rex and Pr, which provides a guidance for the experimental study.

6. Regression Analysis Assume f5 is of an exponential form: where m and n are constants to be determined by the experimental data For a particular fluid, Prn is constant. Then the above equation can be written as Taking Log (Logarithmic function with base 10) on both sides of the equation to obtain: The above equation can be written as Y= a = LogC and b = m, X = a and b can be determined by the Method of Least Square or regression analysis. The residual sum of squares (RSS), also known as the sum of squared residuals (SSR) or the sum of squared errors of prediction (SSE), is the sum of the squares of residuals (deviations predicted from actual empirical values of data). ... A small RSS indicates a tight fit of the model to the data. The coefficient of determination (denoted by R2) is a key output of regression analysis. It is interpreted as the proportion of the variance in the dependent variable that is predictable from the independent variable. ... An R2 between 0 and 1 indicates the extent to which the dependent variable is predictable.

7. Heat Transfer Correlations - Flat Plate in Parallel Flow Uniform surface temperature: Laminar flow (similarity solution): For fluids of small Prandtl number, such as liquid metals, the above equation for heat transfer does not apply,

8. Heat Transfer Correlations - Flat Plate in Parallel Flow = Peclet number Turbulent flow: Mixed boundary layer conditions: If a representative transition Reynolds number of is assumed, the above equation reduces to:

9. Heat Transfer Correlations - Flat Plate in Parallel Flow Uniform surface heat flux: Laminar flow: Turbulent flow:

10. The cylinder in cross flow Flow consideration Bernoulli Relation for Potential Flow Also: Favorable pressure gradient: Adverse pressure gradient:

11. The cylinder in cross flow The Reynolds number: Before the boundary layer separation, the boundary layer may be laminar or turbulent. If , the boundary layer would remain laminar, and separation would occur at If the boundary layer is turbulent. Since the momentum of fluid in a turbulent boundary layer is larger than in the laminar boundary layer, it is reasonable to expect transition to delay the occurrence of separation:

12. The cylinder in cross flow Drag coefficient CD FD – drag force (surface shear stress + pressure differential in the flow direction resulting from formation of the wake) Af – the cylinder frontal area ( the area projected perpendicular to the free stream velocity) Convection Heat Transfer: However, average heat transfer coefficient is considered in this Chapter.

13. Heat Transfer Correlations - Cross Flow Correlation from Hilpert, (7.52), C and m are listed in Table 7.2, properties evaluated at Tf Table 7.3 for the constants of eq. (7.52) for noncircular cylinders in cross flow of a gas

14. Heat Transfer Correlations The correlation due to Zhukauskas is of the form (7.53) where all properties are evaluated at except Prs, which is evaluated at Ts. Values of C and m are listed in Table 7.4. Churchill and Bernstein have proposed a single comprehensive equation that covers the entire range of ReD for which data are available, as well as a wide range of Pr (7.54), where all properties are evaluated at the film temperature

15. Flow across banks of tubes Heat exchangers: Steam generation in a boiler, air cooling in the coil of an air conditioner, … Tube arrangement: Aligned vs. Staggered