Conduction & Convection

1. Conduction & Convection

2. Quiz 9 – 2014.01.27 A flat furnace wall is constructed with a 4.5-inch layer of refractory brick (k = 0.080 Btu/ft·h·F) backed by a 9-inch layer of common brick (k= 0.800 Btu/ft·h·F) and a 2-inch layer of silica foam (k= 0.032 Btu/ft·h·F). The temperature of the inner face of the wall is 1200F, and that of the outer face is 170F. What is the temperature of the interface between the refractory brick and the common brick? What would be the temperature of the outer face if the silica foam is placed between the two brick layers? TIME IS UP!!!

3. Outline • Conduction Heat Transfer • 2.1. Series/Parallel Resistances • 2.2. Geometric Considerations • Convection Heat Transfer • 3.1. Heat Transfer Coefficient • 3.2. Dimensionless Groups for HTC Estimation

4. Geometric Considerations Heat Conduction Through Concentric Cylinders

5. Geometric Considerations Heat Conduction Through Concentric Cylinders

6. Geometric Considerations Heat Conduction Through Concentric Cylinders

7. Geometric Considerations Heat Conduction Through Hollow Spheres Integrating both sides:

8. Geometric Considerations Heat Conduction Through Hollow Spheres Rearranging:

9. Geometric Considerations Heat Conduction Through Hollow Spheres Define a geometric mean area: …and a geometric mean radius: *Final form

10. Shell Balance Plane Wall/Slab

11. Shell Balance Plane Wall/Slab

12. Shell Balance Plane Wall/Slab

13. Shell Balance Plane Wall/Slab

14. Shell Balance Cylinder

15. Shell Balance Sphere

16. Heat Transfer Coefficient Convection Heat Transfer Where: Q = heat flow rate A = heat transfer area h = heat transfer coefficient Tw = temperature at solid wall Tf = temperature at bulk fluid Useful Conversion:

17. Heat Transfer Coefficient Convection Heat Transfer Where: Q = heat flow rate A = heat transfer area h = heat transfer coefficient Tw = temperature at solid wall Tf = temperature at bulk fluid Driving force Thermal Resistance

18. Heat Transfer Coefficient

19. Dimensionless Groups Thermal conductivities are easy to determine by calorimetric experiments, but heat transfer coefficients require the analysis of transfer mechanisms. Mechanism Ratio Analysis 1. Heat, mass, and momentum transport are described by differential equations of change. Navier-Stokes Eq’n e.g.

20. Dimensionless Groups Thermal conductivities are easy to determine by calorimetric experiments, but heat transfer coefficients require the analysis of transfer mechanisms. Mechanism Ratio Analysis 2. However, these equations are complex and most of the time difficult to solve/integrate.

21. Dimensionless Groups Thermal conductivities are easy to determine by calorimetric experiments, but heat transfer coefficients require the analysis of transfer mechanisms. Mechanism Ratio Analysis 3. But still, valuable information is described in these equations, relating the different forces. Pressure Forces Inertial Forces Viscous Forces

22. Dimensionless Groups Thermal conductivities are easy to determine by calorimetric experiments, but heat transfer coefficients require the analysis of transfer mechanisms. Mechanism Ratio Analysis 4. In the Mechanism Ratio Analysis, solving the equations of change is replaced by empiricism. Viscous Forces Inertial Forces Pressure Forces

23. Dimensionless Groups Thermal conductivities are easy to determine by calorimetric experiments, but heat transfer coefficients require the analysis of transfer mechanisms. Mechanism Ratio Analysis 5. This is done by just taking the ratio of the mechanisms and making them into dimensionless groups. Viscous Forces Pressure Forces Inertial Forces Inertial Forces

24. Dimensionless Groups Reynolds Number, Re – the ratio of inertial to viscous forces. Euler Number, Eu – the ratio of pressure to inertial forces. Viscous Forces Pressure Forces Inertial Forces Inertial Forces

25. Dimensionless Groups Reynolds Number, Re – the ratio of inertial to viscous forces. Euler Number, Eu – the ratio of pressure to inertial forces. “If the phenomenon is so complex that negligible knowledge can be gained from the investigation of the differential equations, then empirical processes are available for evolving dimensionless groupings of the involved variables.” – Foust, 1980

26. Dimensionless Groups Useful dimensionless groups for Heat Transfer: cP = specific heat (J/kg-K) μ= viscosity (Pa-s) D = characteristic length (diameter) (m) k = thermal conductivity (W/m-K) h = heat transfer coefficient (W/m2-K)

27. Dimensionless Groups Correlations for Heat Transfer Coefficients: Dittus-Boelter Equation n = 0.4 when fluid is heated n = 0.3 when fluid is cooled (for forced convection/ turbulent, horizontal tubes) Sieder-Tate Equation (for forced convection/ turbulent, Re > 10000 & 0.5 < Pr < 100)

28. Dimensionless Groups Exercise! An organic liquid enters a 0.834-in. ID horizontal steel tube, 3.5 ft long, at a rate of 5000 lb/hr. You are given that the specific heat, thermal conductivity, and viscosity of the liquid is 0.565 Btu/lb-°F, 0.0647 Btu/hr-ft-°F, and 0.59 lb/ft-hr, respectively. All these properties are assumed constant. If the liquid is being cooled, determine the inside-tube heat transfer coefficient using the Dittus-Boelter Equation.