CONVECTIVE HEAT TRANSFER Mohammad Goharkhah Department of Mechanical Engineering, Sahand Unversity of Technology, Tabriz, Iran
CHAPTER 4 HEAT TRANSFER IN CHANNEL FLOW
BASIC CONCEPTS
BASIC CONCEPTS Laminar vs. turbulent flow. Transition from laminar to turbulent flow takes place when the Reynolds number reaches the transition value. For flow through tubes the experimentally determined transition Reynolds number is: Entrance vs. fully developed region. (1)Entrance region (developing region). It extends from the inlet to the section where the boundary layer thickness reaches the channel center. (2) Fully developed region. This zone follows the entrance region. In general the lengths of the velocity and temperature entrance regions are not identical.
Entrance vs. fully developed- Flow Field Entrance Region (Developing Flow, 0 x L h ) Fully Developed Flow Region (L h x) Core velocity u c increases with axial distance x (u c is not constant). Pressure decreases with axial distance (dp/dx<0) Velocity boundary layer thickness is within tube radius (δ<d/ 2 ). Streamlines are parallel v r = 0. For two dimensional incompressible flow the axial velocity u is invariant with axial distance x. That is u/ x=0.
Entrance vs. fully developed- Temperature Field Entrance Region (Developing Temperature, 0 x L t ). Core temperature T c is uniform equal to inlet temperature (T c =Ti ). Temperature boundary layer thickness is within the tube s radius (δ t <D/ 2). Fully Developed Temperature Region (L t x). Fluid temperature varies radially and axially. Thus T / x 0. A dimensionless temperature φ(to be defined later) is invariant with axial distance x. That is φ / x =0.
Determination of Hydrodynamic and Thermal Entrance Lengths- Scale Analysis Hydrodynamic Entrance Length Result of scale analysis for the velocity boundary layer thickness for external flow For the flow through a tube at the end of the entrance region with the Reynolds number based on tube diameter D
Determination of Hydrodynamic and Thermal Entrance Lengths- Scale Analysis
Determination of Hydrodynamic and Thermal Entrance Lengths- Analytic and Numerical Solutions Laminar Flow channel flow area channel perimeter Hydrodynamic Entrance Length Thermal Entrance Length
Determination of Hydrodynamic and Thermal Entrance Lengths- Analytic and Numerical Solutions For a rectangular channel of aspect ratio 2 scaling estimates the constant 0.29 to be unity. For a rectangular channel of aspect ratio 2 at uniform surface temperature scaling estimates the constant 0.22 to be unity.
Determination of Hydrodynamic and Thermal Entrance Lengths- Analytic and Numerical Solutions Turbulent Flow
Determination of Hydrodynamic and Thermal Entrance Lengths- Integral Method The integral momentum and energy equation Since the core flow is inviscid, dp/dx is related to U c (x) through the Bernoulli equation ρu c2 /2 + P = constant
Determination of Hydrodynamic and Thermal Entrance Lengths- Integral Method 1 mass conservation in the channel of half-width (from y = 0 to y = D/2) 2 Solve Eqs. 1 and 2 for δ(x) and U c (x) by first assuming a boundary layer profile shape. Take u/u c = 2y/δ (y/δ) 2 At the location X where the two boundary layers merge we set δ(x) = D/2,
ANALYTICAL SOLUTION OF THE FLOW FIELD
FULLY DEVELOPED FLOW steady-state mass and momentum conservation
FULLY DEVELOPED FLOW- Parallel Plate Channel Conservation of Mass The fully developed region is that section of the duct flow that is situated far enough from the entrance such that the scale of v is negligible Conservation of Momentum y momentum equation P is a function of x only X momentum equation no-slip conditions y is measured away from the centerline of the channel
FULLY DEVELOPED FLOW- Round Tube The momentum equation for a duct of arbitrary cross section the fully developed laminar flow in a round tube of radius r 0 Hagen and Poiseuille solution
HYDRAULIC DIAMETER AND PRESSURE DROP Objective Calculation of the pressure drop in a duct with prescribed flow rate or the calculation of the flow rate in a duct with prescribed pressure drop the momentum theorem in the longitudinal direction Perimeter of the cross section Friction factor The calculation of P is possible provided that we know the friction factor f.
HYDRAULIC DIAMETER AND PRESSURE DROP The friction factors derived from the Hagen Poiseuille solutions: Poiseuille number The product f Re Dh is a number that depends only on the shape of the cross section. This number has been named the Poiseuille number, Po = f Re Dh The fact that f Re Dh is a constant (of order close to 1) expresses the balance between the only two forces that are present, imposed pressure difference and fluid friction.
Calculation of the Friction Factor- Duct of rectangular cross section In general, the friction factor f is obtained by solving the Poisson equation in the duct cross section of interest. Fully developed laminar flow through a duct of rectangular cross section Above Equation can be solved for u(y, z) by Fourier series. Here, we outline a more direct, approximate approach. To calculate f or τ w, we need the velocity distribution u(y, z): From the parallel-plate and round-tube solutions discussed previously, we expect u(y, z) to be adequately represented by the expression The problem reduces to calculating u 0 (the centerline velocity)
Calculation of the Friction Factor- Duct of rectangular cross section Mean Velocity
The present f Re Dh result coincides with the numerical result in the tall and flat cross-sectional shape limits because in those limits the profile shape assumption is exactly the Hagen Poiseuille profile shape. Overall, the agreement between the current result and numerically derived results is better than 15 percent.
HEAT TRANSFER TO FULLY DEVELOPED DUCT FLOW
HEAT TRANSFER TO FULLY DEVELOPED DUCT FLOW Mean Temperature The first law of thermodynamics If the fluid as an ideal gas If the fluid is an incompressible liquid with negligible pressure changes (dh = c p dt m ) (dh = c dt m )
Fully Developed Temperature Profile Objective Calculation of the convective heat transfer coefficient, h How??? We must first determine the temperature field in the fluid T(x, r) by solving the energy equation subject to appropriate wall temperature boundary conditions. The energy equation for steady, ɵ-symmetric flow through a round tube In the hydrodynamic fully developed region v = 0 and u = u(r) balance among three possible energy flows
Fully Developed Temperature Profile Multiplying scales by D 2 /T and using the definition of heat transfer coefficient h = q /ΔT, we obtain: The longitudinal conduction effect is negligible if: from the convection radial conduction balance, we learn that the Nusselt number is a constant of order 1
Fully Developed Temperature Profile Assumptions The flow is hydrodynamically fully developed; hence, the velocity profile u(r) is the same at any x along the duct. we assumed that the scale of 2 T/ r 2 is T/D 2 ; in other words, the effect of thermal diffusion has had time to reach the centerline of the stream. This last assumption is not valid in a thermal entrance region X T near the duct entrance, where the proper scale of 2 T/ r 2 is T/δ T2, with δ T <<D.
Definition of Fully Developed Temperature Profile
Uniform Wall Heat Flux Fully developed temperature differentiate with respect to x This means that the temperature everywhere in the cross section varies linearly in x
Uniform Wall Heat Flux Fully developed temperature profile in a round tube with uniform heat flux
Hagen Poiseuille velocity profile Energy equation Integrating this equation twice and invoking one boundary condition (finite φ at r * = 0) C2 is obtained from The mean temperature difference
Uniform Wall Temperature A round tube with wall temperature T 0 independent of x The stream bulk temperature is T 1 at some place x = x 1 in the fully developed region Eliminating q (x) and integrating the result from T m = T 1 at x = x1 yields
Uniform Wall Temperature Hagen Poiseuille velocity profile Energy equation T = T 0 φ(t 0 T m ) boundary conditions
Uniform Wall Temperature * boundary conditions + The radial profile φ will be a function of both r * and Nu, where Nu is the unknown in this problem. The additional condition for determining Nu uniquely is the definition of the heat transfer coefficient ** The problem statement is now complete: The value of Nu must be such that the φ(r*, Nu) solution of Eqs. (*) and (+) satisfies the Nu definition (**).
Uniform Wall Temperature The actual solution may be pursued in a number of ways, for example, by successively approximating (guessing) and improving the φ solution. It is more convenient to solve the problem numerically. the differential energy equation is first approximated by finite differences and integrated from r * = 1 to r * = 0. To perform the integration at all, we must first guess the value of Nu, which also gives us a guess for the initial slope of the ensuing φ(r * ) curve. The success of the Nu guess is judged by means of the first of boundary conditions. the refined result is ultimately
Uniform Wall Temperature (*) and (+) (*) (*) 2 1
Uniform Wall Temperature 1 2
Uniform Wall Temperature
Example Air flows with a mean velocity of 2 m/s through a tube of diameter 1.0 cm. The mean temperature at a given section in the fully developed region is 35oC. The surface of the tube is maintained at a uniform temperature of 130oC. Determine the length of the tube section needed to raise the mean temperature to 105oC. To compute L, it is necessary to determine the properties and the average h. Air properties are determined at the mean temperature, defined as
The slight variation in these values mimics that of B = (πd 2 h/4)/aduct, implying that it is caused by hydraulic diameter nondimensionalization, that is, by the mismatch between hydraulic diameter and effective wall stream distance. This behavior again stresses the importance of the new dimensionless group B = (πd 2 h/4)/aduct.
the fully developed values of the friction factor and the Nusselt number in a duct with regular polygonal cross section The thermally developed Nu value is considerably smaller in fully developed flow than in slug flow. The latter refers to the flow of a solid material (slug, rod), or a fluid with an extremely small Prandtl number (Pr 0), where the viscosity is so much smaller than the thermal diffusivity that the longitudinal velocity profile remains uniform over the cross section, u = U, like the velocity distribution of a solid slug.
HEAT TRANSFER TO DEVELOPING FLOW Thermally Developing Hagen Poiseuille Flow the velocity profile is fully developed while the temperature profile is just being developed Neglecting the effect of axial conduction (Pe x >> 1), Isothermal entering fluid, T = T IN for x < 0, where x is measured (positive) downstream from the location X
HEAT TRANSFER TO DEVELOPING FLOW This problem was treated for the first time by Graetz Graetz problem The energy equation is linear and homogeneous. Separation of variables is achieved by assuming a product solution for ɵ * (r *, x * ), Sturm Liouville type Cn Constants determined by the x* = 0 condition
HEAT TRANSFER TO DEVELOPING FLOW The average heat transfer coefficient is obtained by integrating the local heat transfer coefficient along a tube of length.
HEAT TRANSFER TO DEVELOPING FLOW A simpler approach The overall Nusselt number for the thermal entrance region of a tube with isothermal wall is defined as
HEAT TRANSFER TO DEVELOPING FLOW
HEAT TRANSFER TO DEVELOPING FLOW
HEAT TRANSFER TO DEVELOPING FLOW
solution HEAT TRANSFER TO DEVELOPING FLOW- Leveque solution Simpler alternative to the Graetz series solution, which is known as the L evˆeque solution
Thermally developing Hagen Poiseuille flow in a round tube with uniform heat flux
Thermally developing Hagen Poiseuille flow in a in a parallel-plate channel with isothermal surfaces heat transfer to thermally developing Hagen Poiseuille flow in a parallel-plate channel with isothermal surfaces are approximated by
Thermally developing Hagen Poiseuille flow in a in a parallel-plate channel with uniform heat flux
Thermally and Hydraulically Developing Flow The most realistic (and most difficult) tube flow problem consists of solving the following equation with the Hagen Poiseuille profile 2(1 r * 2 ) replaced by the actual x-dependent velocity profile present in the hydrodynamic entry region. This, the finite-pr problem, has been solved numerically by a number of investigators. The analytical expressions recommended for the local and overall Nusselt numbers in the range 0.1 < Pr < 1000 in parallel-plate channels are:
Thermally and Hydraulically Developing Flow
A general expression for the entrance and fully developed regions A closed-form expression for the local Nusselt number that covers both the entrance and fully developed regions in a tube with uniform heat flux is