Estimation of Convective Heat Transfer Coefficient

Convective heat transfer coefficient Convective heat transfer coefficient (h) is predicted from empirical correlations. The coefficient is influenced by such parameters as type and velocity of fluid, physical properties of fluid, temperature difference, and the geometrical shape of the physical system underconsideration. Dimensional analysis is used to develop empirical correlations that allow estimation of h. Following correlations apply to Newtonian fluids only. For expression for non-Newtonian fluids, the textbook by Heldman and Singh (1981) is recommended.

Force convection Fluid is forced to move over an object by external mechanical means NNu = (NRe, NPr) Where NNu = Nusselt number = hD/k h = convective heat-transfer coefficient (W/ m2oC) D = characteristic dimension (m) k = thermal conductivity (W/moC) NRe = Reynolds number = uD/ NPr = Prandtl number = Cp/k

Laminar flow in horizontal pipes When Reynolds number < 2100 D L For (NRe x NPr x )  100 D L For (NRe x NPr x ) > 100 All physical properties are evaluated at bulk fluid temp. except w is at surface temp of wall. D = characteristic dimension = diameter of pipe

Example Water flowing at a rate of 0.02 kg/s is heated from 20 to 60C in a horizontal pipe (inside diameter = 2.5 cm). The inside pipe surface temperature is 90C. Estimate h if the pipe is 1 m long.

Transition flow in horizontal pipes When Reynolds number between 2100 and 10000

Turbulent flow in horizontal pipe When Reynolds number > 10000 All physical properties are evaluated at bulk fluid temp. except w is at surface temp of wall. D = characteristic dimension = diameter of pipe

Example Water flowing at a rate of 0.2 kg/s is heated from 20 to 60C in a horizontal pipe (inside diameter = 2.5 cm). The inside pipe surface temperature is 90C. Estimate h if the pipe is 1 m long.

Example What is the expected percent increase in convective heat transfer coefficient if the velocity of the fluid is doubled while all other related parameters are kept the same for turbulent flow in a pipe.

Convection in non-circular ducts Equations for circular tube with hydraulic diameter

Flow past immersed objects For flat plate For cylinder if fluid is gas NNu = C NRe if fluid is liquid NNu = C NRenNPr1/3 NNu = 0.664 NRe NPr 1/3 1/2 NRe C n 0.4-4 4-40 40-4000 4000-40000 40000-400000 0.989 0.911 0.683 0.193 0.0266 0.330 0.385 0.466 0.618 0.805

Flow past immersed objects

Flow past immersed objects For single sphere NNu = 2 + 0.60NRe0.5 X NPr1/3 where 1 < NRe < 70,000 0.6 < NPr < 400 Fluid properties are evaluated at film temperature (Tf) where Tf = (Twall + Tmedium) / 2

Example Calculate convective heat transfer coefficient when air at 90C is passed through a deep bed of green peas. Assume surface temperature of a pea to be 30C. The diameter of each pea is 0.5 cm. The velocity of air through the bed is 0.3 m/s.

Free convection Free convection occurs due to density differences in fluids as they come into contact with a heated surface. The low density of fluid at a higher temperature causes buoyancy forces, and as a result, heated fluid moves upward and colder fluid takes its place

hD k NNu = = a (NGr NPr)m where a, m = constants NGr = Grashof number = D32gT/2 D = characteristic dimension (m)  = coefficient of volumetric expansion (K-1) T = Temperature difference between wall and surrounding bulk (oC) All physical properties are evaluated at film temperature (Tf = (Tw+Tb)/2)

Use Figure A Use Figure B

Figure A

Figure B

Example Estimate the convective heat transfer coefficient for convective heat loss from a horizontal 10 cm diameter stem pipe. The surface temperature of the uninsulated pipe is 130C, and the air temperature is 30C

Other empirical equations for h estimation

1. Forced Convection Flow Inside a Circular Tube All properties at fluid bulk mean temperature (arithmetic mean of inlet and outlet temperature). Nusselt numbers Nu0 from sections 1-1 to 1-3 have to be corrected for temperature-dependent fluid properties according to section 1-4.

Constant wall temperature: (Hausen) 1-1 Thermally developing, hydrodynamically developed laminar flow (Re < 2300) Constant wall temperature: (Hausen) Constant wall heat flux: (Shah)

1-2 Simultaneously developing laminar flow (Re < 2300) Constant wall temperature: (Stephan) Constant wall heat flux: which is valid over the range 0.7 < Pr < 7 or if Re Pr D/L < 33 also for Pr > 7.

1-3 Fully developed turbulent and transition flow (Re > 2300) Constant wall heat flux: (Petukhov, Gnielinski) where Constant wall temperature: For fluids with Pr > 0.7 correlation for constant wall heat flux can be used with negligible error.

1-4 Effects of property variation with temperature Liquids, laminar and turbulent flow: Subscript w: at wall temperature, without subscript: at mean fluid temperature Gases, laminar flow: Nu = Nu0 Gases, turbulent flow: Temperatures in Kelvin

2. Forced Convection Flow Inside Concentric Annular Ducts, Turbulent (Re > 2300) All properties at fluid bulk mean temperature (arithmetic mean of inlet and outlet temperature). Dh = Do - Di                                  

Heat transfer at the inner wall, outer wall insulated: (Petukhov and Roizen) Heat transfer at the outer wall, inner wall insulated: Heat transfer at both walls, same wall temperatures: (Stephan)

Equations for circular tube with hydraulic diameter 3. Forced Convection Flow Inside Non-Circular Ducts, Turbulent (Re > 2300) Equations for circular tube with hydraulic diameter

4. Forced Convection Flow Across Single Circular Cylinders D = cylinder diameter, um = free-stream velocity, all properties at fluid bulk mean temperature.

4-1 Smooth circular cylinder Valid over the ranges 10 < Rel < 107 and 0.6 < Pr < 1000                                         (Gnielinski) where                                                                                   

4-2 Effects of property variation with temperature Liquids:                                    Subscript w: at wall temperature, without subscript: at mean fluid temperature. Gases:                           Temperatures in Kelvin.

5. Forced Convection Flow over a Flat Plate                   All properties at mean film temperature                

Laminar boundary layer, constant wall temperature: (Pohlhausen) valid for ReL < 2x105, 0.6 < Pr < 10 Turbulent boundary layer along the whole plate, constant wall temperature:                                               (Petukhov) Boundary layer with laminar-turbulent transition:                                  (Gnielinski)

L = characteristic length 6. Natural Convection All properties at                                                  L = characteristic length

  Nu0 "Length" L Vertical wall 0.67 H Horizontal cylinder 0.36 D Sphere 2.00

For ideal gases:       (temperature in K)                                                (Churchill, Thelen) valid for 10-4 < Gr Pr < 4x1014, 0.022 < Pr < 7640, and constant wall temperature

Combined free and forced convection From J.P. Holman (1992)

UWT = uniform wall temp., UHF = uniform heat flux Aiding flow = forced and free convec. Are in the same direction while opposite flow means they are in opposite direction.

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