# HW #4 /Tutorial # 4 WRF Chapter 18; HW #4 /Tutorial # 4 WRF Chapter 18; WWWR Chapter 19 ID Chapter 6 Homework # 4 (Self

Tutorial # 4 practice) WWWR #19.1,19.4, WRF#18.1, 18.3 ID # WRF#18.10.

6.35, 6.36, 6.39 ID # 6.33. 6.37. To be discussed during # Correction the week 10-13 Feb., Question 19.19

2020. By either volunteer or 3.94 Pr (Pr 0.954) Gr x class list. 1/ 2

1/ 4 1/ 4

x Nu x 0.508Pr1/ 2 (Pr 0.954) 1/ 4 Grx1/ 4 ; hx Nu x k

x Convective Heat Transfer Fundamental Considerations In

Convective Heat Transfer Two main classifications of convective heat transfer These have to do with the driving force causing fluid to flow

Natural or free convection Fluid motion results from heat transfer Fluid heated/ cooled -> density change/ buoyancy effect -> natural circulation in which affected fluid moves of its own accord past the solid surface

- fluid it replaces is similarly affected by the energy transfer process is repeated Forced convection Fluid circulation is produced by external agency (fan or pump) Analytical Methods

(a) Dimensional Analysis (b) Exact Analysis of the Boundary Layer (c) Approximate Integral Analysis (d) Analogy between Energy and Momentum Exchange

Significant Parameters In Convective Heat Transfer A. Both have same dimensions L2/t; thus their ratio must be dimensionless

B. This ratio, that of molecular diffusivity of momentum to the molecular diffusivity of heat, is designed the Prandtl number Pr

cp = k Prandtl number

observed to be a combination of fluid properties; thus Pr itself may be thought of as a property. Primarily a function of temperature s

A ratio of conductive thermal resistance to the convective thermal resistance of the fluid Nusselt number Nu hL

k Where the thermal conductivity of the fluid as opposed to that of the solid, which was the case in the evaluation of the Biot modulus. Dimensional Analysis of

Convective Energy Transfer Forced Convection Dimensional Analysis for Forced

Convection Natural Convection Courtesy Contribution by ChBE Year

Representative, 2004. Dimensional Analysis for Natural Convection

Exact Analysis of the Laminar Boundary Layer A logical consequence of this situation is that the hydrodynamic and thermal boundary layers are of equal thickness.

It is significant that the Prandtl numbers for most gases are sufficiently closed to unity that the hydrodynamic and thermal boundaries are of similar extent. See detailed derivation below

Detailed Derivation Eq.(19-19) WWWR page 295 T

y y 0

T y

T T S f 0

2 y

0.664 T TS

y Re x 2x Re x

2X From Eq.(19-18) 0.332 T TS

Re x X 1

0.332 T TS Re x 2 X

(19-19) Approximate Integral Analysis of

the Thermal Boundary Layer An approximate method for analysis of the thermal boundary layer employs the integral analysis used by von Krmn for the hydrodynamic boundary layer.

Consider the control volume designated by the dashed lines, applying to flow parallel to a flat surface with no pressure gradient, having width x, a height equal to the thickness of the thermal boundary layer

0.323 Energy and Momentum Transfer Analogies

The Colburn analogy expression is St Pr 2/3 = Cf (19-37) 2

8) 9) s on

er Example 1 o Water at 50 F enters a heat-exchanger tube having an inside diameter of 1 in and a length of 10 ft. The water

o flows at 20 gal/min. For a constant wall temperature of 210 F estimate the =(90+210)/2 = 150

Film temperature = (water mean bulk temperature + pipe wall temperature)/2 = (50+130)/2 o

Second iteration is required since if |TL 130| > 3 F?