Lecture 3 - For heat transfer 2 PDF

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For heat transfer 2...

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ME 303 Convection, Boiling, Condensation and Mass Transfer Semester: August 2015 Dr. Sumon Saha Assistant Professor Department of Mechanical Engineering Bangladesh University of Engineering and Technology Thursday, November 12, 2015

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Today’s Topic

Dimensional Analysis

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The equations of convective heat transfer

1. Continuity equation

∂u ∂v ∂w + + =0 ∂x ∂y ∂z

2. Momentum equations  ∂ 2u ∂ 2u ∂ 2u   ∂u ∂u ∂u ∂u  ∂p ρ +u +v +w  = − +µ 2 + 2 + 2 + X ∂x ∂y ∂z  ∂x ∂y ∂z   ∂t  ∂x  ∂ 2v ∂ 2v ∂ 2 v   ∂v ∂v ∂v ∂v  ∂p ρ  + u + v + w  = − + µ  2 + 2 + 2  +Y ∂x ∂y ∂z  ∂y  ∂t  ∂x ∂y ∂z 

 ∂w  ∂2w ∂2w ∂2w  ∂w ∂w ∂w  ∂p ρ +u +v + w  = − +µ 2 + 2 + 2 +Z ∂x ∂y ∂z  ∂z ∂y ∂z   ∂t  ∂x 3. Energy equation  ∂ 2T ∂ 2T ∂ 2T  ∂T ∂T ∂T ∂T  +u +v +w ρC p   = k 2 + 2 + 2 ∂x ∂y ∂z  ∂y ∂z  ∂t  ∂x

  + q′′′ + µΦ 

Three-dimensional, incompressible flow, Newtonian fluid with constant properties in Cartesian coordinate system.

Navier-Stokes and Energy equations

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 Three dependent variables  V = uiˆ + vjˆ + wkˆ 1. Velocity vector 2. Pressure (P) 3. Temperature (T)  Four independent variables (Cartesian coordinate) 1. x 2 y 2. (space) 3. z 4. t (time)  Four known parameters (fluid properties) 1.  (density) 2. µ (dynamic viscosity) 3. k (thermal conductivity) 4. Cp (specific heat at constant pressure)

(

)

All are dimensional quantities and therefore, the total number of inputs are 4.

Non-dimensional or Dimensionless Parameters   Some of the important parameters for convection problem are listed below 1. Reynolds number (Re) 2. Prandtl number (Pr) 3. Grashof number (Gr) Governing parameters 4. Rayleigh number (Ra) 5 Peclet number (Pe) 5. 6. Richardson number (Ri) 7. Nusselt number (Nu) 8. Biot number (Bi) Performance parameters 9. Stanton number (St)

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Reynolds Number

Reynolds number (Re) is the ratio of inertial force to viscous force.

inertia force ρuref Lref uref Lref = = Re = µ ν viscous force

where,  = density of fluid, kg/m3 µ = dynamic viscosity of fluid, Pa.s ν = kinematic viscosity of fluid, m2/s uref = characteristic or reference velocity, m/s Lref = characteristic or reference length, m

Osborne Reynolds

Re can be used as the criterion to determine the change from laminar to turbulent flow. As the Reynolds number increased, the inertia forces become dominant and small disturbances in the fluid may be amplified to cause the transition from laminar to turbulent flow.

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Reynolds Number

Flow over a flat plate

ReL =

u∞ L

ν

> 5 × 105

Flow inside tube, pipe or duct Laminar flow

2300 ≤ ReD =

uavg D

µ

≤ 10000

Transitional flow

Turbulent flow

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Prandtl Number

Prandtl number (Pr) is the ratio of molecular diffusivity of momentum to molecular diffusivity of heat.

molecular diffusivity of momentum µC p ν Pr = = = molecular diffusivity of heat α k where, where ν= kinematic viscosity of fluid, m2/s α = thermal diffusivity of fluid, m2/s

Ludwig Prandtl

It represents the relative importance of momentum and energy transport by the diffusion process. Pr relates the relative thickness of the hydrodynamic and thermal boundary layers. Pr is the connecting link between the velocity field and thermal field.

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Prandtl Number

Variation of fluid properties and Pr of air with temperature at Patm

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Prandtl Number

Variation of fluid properties and Pr of water with temperature

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Prandtl Number

Typical range of Pr for common fluids

Question: Define Prandtl number. Explain its physical significance in relation to (forced and free) convection heat transfer.

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Grashof Number

Grashof number (Gr) is the ratio of the buoyancy force to the viscous force acting on the fluid. 3 buoyancy force g β∆TLref = Gr = viscous force ν2

where, ν = kinematic viscosity of fluid, m2/s g = gravitational acceleration, m/s2 Franz Grashof  = coefficient of volume expansion, 1/K T = temperature difference, K Lref = characteristic or reference length of the geometry, m Gr is a measure of the relative magnitudes of the buoyancy force and the opposing viscous force acting on the fluid. The flow regime in natural convection is governed by Grashof number. It provides the main criterion in determining whether the fluid flow is laminar or turbulent in natural convection.

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Rayleigh Number

Rayleigh number (Ra) is the ratio of the buoyancy forces and (the products of) thermal and momentum diffusivities.

Ra =

g β∆TL3ref

= Gr Pr

να where, ν = kinematic viscosity of fluid, m2/s  = thermal diffusivity of fluid, m2/s Lord Rayleigh g = gravitational acceleration, m/s2  = coefficient of volume expansion, 1/K T = temperature difference, K Lref = characteristic or reference length of the geometry, m When the Rayleigh number is below a critical value for that fluid, heat transfer is primarily in the form of conduction; when it exceeds the critical value, heat transfer is primarily in the form of convection.

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Peclet Number

Peclet number (Pe) is the ratio of the rate of advection of a physical quantity by the flow to the rate of diffusion of the same quantity driven by an appropriate gradient

rate of advection u ref Lref = = ReL Pr Pe = rate of diffusion α where, Jean Claude Eugene Peclet  = thermal diffusivity of fluid, m2/s uref = characteristic or reference velocity, m/s Lref = characteristic or reference length of the geometry, m Peclet number is a dimensionless number used in calculations involving convective heat transfer. If Pe is small, conduction is important and in such a case, the major source of conduction could be down the walls of the solid surface.

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Richardson Number

Richardson number (Ri) is the ratio of the buoyancy term to the flow gradient term.

g β∆TLref Gr buoyancy term = = 2 Ri = flow gradient term u ref Re where, g = gravitational acceleration, acceleration m/s2  = coefficient of volume expansion, 1/K Lewis Fry Richardson T = temperature difference, K uref = characteristic or reference velocity, m/s Lref = characteristic or reference length of the geometry, m When Ri >>1, inertia forces are negligible and natural convection effects dominate. When Ri...