FREE CONVECTION Nazaruddin Sinaga Laboratorium Efisiensi dan Konservasi Energi Jurusan Teknik Mesin Universitas Diponegoro FREE CONVECTION The cooling of a boiled egg in a cooler environment by natural convection. The warming up of a cold drink in a warmer environment by natural convection Natural Convection Where weve been Up to now, have considered forced convection, that is

an external driving force causes the flow. Where were going: Consider the case where fluid movement is by buoyancy effects caused by temperature differential Events due to natural convection Weather events such as a thunderstorm Glider planes Radiator heaters Hot air balloon Heat flow through and on outside of a double pane window

Oceanic and atmospheric motions Coffee cup example . Small velocity Natural Convection New terms Volumetric thermal expansion coefficient Grashof number Rayleigh number Buoyancy is the driving force Stable versus unstable conditions Nusselt number relationship for laminar free convection on hot or cold surface Boundary layer impacts: laminar turbulent Buoyancy is the driving force in Natural Convection

Buoyancy is due to combination of Differences in fluid density Body force proportional to density Body forces namely, gravity, also Coriolis force in atmosphere and oceans Convection flow is driven by buoyancy in unstable conditions Fluid motion may be (no constraining surface) or along a surface Buoyancy is the driving force Free boundary layer flow Heated wire or hot pipe A heated vertical plate Typical velocity and temperature profiles for

natural convection flow over a hot vertical plate at Ts inserted in a fluid at temperature T. Natural Convection Boundary Layer : Governing Equations The difference between the two flows (forced flow and free flow) is that, in free convection, a major role is played by buoyancy forces. X g Very important Consider the x-momentum equation. u

u 1 P 2u u v g 2 x y x y As we know,p / y 0 , hence the x-pressure gradient in the boundary layer must equal that in the quiescent region outside the boundary layer. Pascal Law : P

- g x u u 1 2u u v g g 2 x y y 2 u

u u 2 u v g x y y Buoyancy force Governing Equations Define , the volumetric thermal expansion coefficient. 1

T P For all liquids and gases 1 1 T T T (T T ) Density gradient is due to the temperature gradient For an ideal gas : P 1

Thus : T RT P RT Governing Equations Buoyancy effects replace pressure gradient in the momentum equation. u u 2u

u v g (T T ) v 2 x y y The buoyancy effects are confined to the momentum equation, so the mass and energy equations are the same. u v 0 x y 2 T T T u u v

2 x y y c p y 2 Strongly coupled and must be solved simultaneously Dimensionless Similarity Parameter x x L u u

u0 and and y y L v v u0 T T T Ts T

* where L is a characteri stic length, and u 0 is an arbitrary reference velocity The x-momentum and energy equations are * * 2 * g ( T T ) L u

u 1 u * s u * * v* * T x y u02 Re L y * 2 * * 2 *

T T 1 T u * * v* * x y Re L Pr y * 2 Dimensionless Similarity Parameter Define new dimensionless parameter, 2 3

g (Ts T ) L u0 L g (Ts T ) L GrL 2 2 u0 Grashof number in natural convection is analogous to the Reynolds number in forced convection. Grashof number indicates the ratio of the buoyancy force to the viscous force.

Higher Gr number means increased natural convection flow GrL 1 forced 2 Re L GrL 1 2 Re L natural for vertical flat plates for pipes

for bluff bodies The transition to turbulent flow occurs in the range for natural convection from vertical flat plates. At higher Grashof numbers, the boundary layer is turbulent; at lower Grashof numbers, the boundary layer is laminar. where the L and D subscripts indicates the length scale basis for the Grashof Number. g = acceleration due to Earth's gravity = volumetric thermal expansion coefficient (equal to approximately 1/T, for ideal fluids, where T is absolute temperature) Ts = surface temperature T = bulk temperature L = length D = diameter = kinematic viscosity

Franz Grashof Born 11 July 1826 Dsseldorf, Germany Died 26 October 1893 (aged 67) Karlsruhe, Germany Nationality German Fields Engineering Laminar Free Convection on Vertical Surface As y : u = 0, T = T

As y 0 : u = 0, T = Ts Ts T u(x,y) Ts With little or no external driving flow, Re 0 and forced convection effects can be safely neglects T g x y u v

GrL 1 2 Re L Nu L f (GrL , Pr) The simple empirical correlations for the average Nusselt number Nu in natural convection are of the form : where RaL is the Rayleigh number, which is the product of the Grashof and Prandtl numbers: o The values of the constants C and n depend on the geometry of the surface and the flow regime, which is characterized by the range of the Rayleigh number. o The value of n is usually for laminar flow and 1/3 for turbulent

flow, while the value of the constant C is normally less than 1. o All fluid properties are to be evaluated at the film temperature Tf = (Ts + T)/2. Empirical solution for the local Nusselt number in laminar free convection hx GrL Nu x k 4 1/ 4 f (Pr) Where f Pr

0.75 Pr 0.609 1.221 Average Nusselt # = Pr 1.238 Pr 1/ 4 h L 4 GrL NuL k

3 4 1/ 4 f (Pr) Effects of Turbulence Just like in forced convection flow, hydrodynamic instabilities may result in the flow. For example, illustrated for a heated vertical surface: Define the Rayleigh number for relative magnitude of buoyancy and viscous forces Rax ,c Grx ,c Pr g (Ts T ) x

Ts T 3 Empirical Correlations Typical correlations for heat transfer coefficient developed from experimental data are expressed as: hL Nu L CRaLn k g Ts T L3 RaL GrL Pr n 1 / 4

n 1 / 3 For Turbulent For Laminar Vertical Plate at constant Ts Log10 Nu L Log10 Ra L Alternative applicable to entire Rayleigh number range (for constant Ts) 0.387 Ra1L/ 6 Nu L 0.825

1 (0.492 / Pr) 9 /16 Vertical Cylinders 8 / 27 2 DD Use same correlations for vertical flat plate if:

D ~ 35 1/ 4 L GrL L Free Convection : Vertical Plate Hot plate or Cold fluid Cold plate or Hot fluid Free Convection from Inclined Plate Cold plate or Hot fluid Hot plate or Cold fluid Horizontal Plate

Cold Plate (Ts < T) Hot Plate (Ts > T) Active Upper Surface Active Lower Surface Empirical Correlations : Horizontal Plate Define the characteristic length, L as As L P Upper surface of heated plate, or Lower surface of cooled plate : Nu L 0.54 Ra1L/ 4

1/ 3 Nu L 0.15 Ra L 104 Ra L 107 7 11 10 Ra L 10 Lower surface of heated plate, or Upper surface of cooled plate : Nu L 0.27 1/ 4 Ra L 10 5

10 Ra L 10 Note: Use fluid properties at the film temperature Ts T Tf 2 Empirical Correlations : Long Horizontal Cylinder Very common geometry (pipes, wires) For isothermal cylinder surface, use general form equation for computing Nusselt #

hD NuD CRaDn k Constants for general Nusselt number Equation RaD 10 10 - 10 2 10 2 - 10 2 C

n 0.675 0.058 1.02 0.148 2 4 0.850 0.188

4 7 0.480 0.250 7 12 0.125 0.333 10 - 10

10 - 10 10 - 10 The End Terima kasih EFFICIENCY AND ENERGY CONSERVATION LABORATORY OF DIPONEGORO UNIVERSITY 36