Convection Part1 External Flow

Introduction Recall: Convention is the heat transfer mode between a fluid and a solid or a 2 fluids of different phases In order to simplify the process we used Newton’s correlation Where h is the convective heat transfer coefficient also called the film coefficient. h is a function of:Fluid flow Fluid properties Geometry of the solid

There are four means to evaluate the heat transfer coefficient 1)Dimensional analysis 2)Exact analysis of boundary layer 3)Approximate integral analysis of the boundary layer 4) Analogy between energy and momentum transfer Significant Parameters: Nusselt Number Nu x y

The heat transfer rate between the surface and the fluid is At the surface itself Where k is the thermal conductivity of the fluid. Therefore:

Momentum Diffusivity Thermal Diffusivity The ratio of the momentum diffusivity over the thermal diffusivity is a combination of fluid properties and is also thougth of as a property (Named Prandtl Number Pr). Dependent on fluid and temperature Prandtl Number Pr

Dimensional Analysis of Convective Heat Transfer Forced Convection:movement dictated by v VariableSymbolDimensions Tube DiameterDL Fluid densityρM L -3 Fluid viscosityμM L -1 t -1 Fluid heat capacityCpQ M –1 T –1 Fluid thermal conductivitykQ t –1 L –1 T –1 VelocityvL t –1 Heat transfer coefficienthQ t –1 L –2 T –1

Using the Buckingham method we group the variables in dimensionless number: This dimensional analysis for a forced convection in a circular conduit indicates the possibility of correlating the variables as Similarly we could have developed the Stanton number instead of the Nusselt

Free Convection:movement dictated by buoyancy Given the coefficient of thermal expansion β: VariableSymbolDimensions Significant lengthDL Fluid densityρM L -3 Fluid viscosityμM L -1 t -1 Fluid heat capacityCpQ M –1 T –1 Fluid thermal conductivitykQ t –1 L –1 T –1 Fluid Coef. Therm. Exp.βT –1 Gravitational accelerationGL t –2 Temperature differenceΔTΔTT Heat transfer coefficienthQ t –1 L –2 T –1

Using the Buckingham method we group the variables in dimensionless number: Define the Grashof number as This dimensional analysis for a forced convection in a circular conduit indicates the possibility of correlating the variables as

GroupSymbolDefinitionInterpretation Grashof NumberGrRatio buoyancy to viscous forces Colburn FactorjHjH Dimensionless heat transfer coefficient Nusselt NumberNuDimensionless surface temperature gradient Prandtl NumberPrRatio momentum to thermal diffusivity ReynoldsReRatio inertia to viscous forces Stanton NumberStModified Nusselt number Peclet NumberPeRePrIndependent heat transfer parameter Selected Dimensionless Groups

Flat Plate in Parallel Flow Laminar Flow Turbulent Flow Transition Region δ(x) x L Properties of fluid evaluated at the film temperature T f

Forced Convection Flat Plate in Parallel Flow Laminar flow:Re<2 x 10 5 Prandtl number >0.6 The local Nusselt number is The average Nusselt number All Prandtl number and Pe >100 The local Nusselt number is The average Nusselt number x L

Forced Convection Flat Plate in Parallel Flow Transition flow: Re c =5 x >Prandtl number >0.6 3 x 10 6 >Re > 2 x 10 5 The average Nusselt number L

Forced Convection Flat Plate in Parallel Flow Turbulent flow: Re>3x >Prandtl number > >Re >3 x 10 6 The average Nusselt number The local Nusselt number L

Cylinder in a Cross Flow Separation v D Transition Laminar Turbulent D Separation v Properties of fluid evaluated at the film temperature T f

Forced Convection Cylinder in a Cross Flow The average Nusselt number If Re D Pr>0.2 Re D Cm , , ,

Forced Convection Various Object in a Cross Flow The average Nusselt number GeometryRe D Cm Square 5x Square 5x Hexagon 5x x x Hexagon 5x Vertical Plate 4x x D D D D D

Sphere in a Cross Flow All properties of fluid evaluated at temperature, except μ s at T s Restrictions 0.71 < Pr < < Re D < 7.6x10 4

Bank of Tubes in a Cross Flow V Fluid in cross flow over tube bank

Aligned Bank of Tubes in a Cross Flow STST D A1A1 SLSL Properties of fluid evaluated at the film temperature T f

Staggered Bank of Tubes in a Cross Flow STST A1A1 Properties of fluid evaluated at the film temperature T f If else SLSL D

Number of row (N L ) greater or equal to < Re D,max < Pr > 0.7 C 1 in table 7.5 If number of row is smaller than 10 C 2 in table 7.6

Number of row (N L ) greater or equal to < Re D,max < 2x > Pr > 0.7 C in table 7.7 If number of row is smaller than 10 C 2 in table 7.8 All properties of fluid evaluated at the average temperature except Pr s at T s

In this case the temperature difference in the convective heat transfer equation is defined as the log-mean temperature difference ΔT lm Where T i is the temperature of the fluid entering the bank T o is the temperature of the fluid leaving the bank And the outlet temperature can be estimated using Where N is the total number of tube and N T the transverse number of tube. Finally the heat transfer rate per unit length is

Packed Bed Properties of fluid evaluated at the the average temperature ε is the porosity or void fraction of the bed (0.3 to 0.5) Valid for gas flow

A p,T is the total area of the particles and A b,c is the bed cross sectional area