ME 259 Heat Transfer Lecture Slides III Dr. Gregory A. Kallio Dept. of Mechanical Engineering, Mechatronic Engineering & Manufacturing Technology California State University, Chico 3/7/05 ME 259

Reading: Incropera & DeWitt Chapter 5 Transient Conduction Reading: Incropera & DeWitt Chapter 5 3/7/05 ME 259

Introduction Transient = Unsteady (time-dependent ) Examples Very short time scale: hot wire anemometry (< 1 ms) Short time scale: quenching of metallic parts (seconds) Intermediate time scale: baking cookies (minutes) Long time scale – daily heating/cooling of atmosphere (hours) Very long time scale: seasonal heating/cooling of the earth’s surface (months) 3/7/05 ME 259

Governing Heat Conduction Equation Assuming k = constant, For 1-D conduction (x) and no internal heat generation, Solution, T(x,t) , requires two BCs and an initial condition 3/7/05 ME 259

The Biot Number The Biot number is a dimensionless parameter that indicates the relative importance of of conduction and convection heat transfer processes: Practical implications of Bi << 1: objects may heat/cool in an isothermal manner if they are small and metallic they are cooled/heated by natural convection in a gas (e.g., air) 3/7/05 ME 259

The Lumped Capacitance Method (LCM) LCM is a viable approach for solving transient conduction problems when Bi<<1 Treat system as an isothermal, homogeneous mass (V) with uniform specific heat (cp) For suddenly imposed, uniform convection (h, T) boundary conditions, solution yields: where 3/7/05 ME 259

General LCM LCM can be applied to systems with convection, radiation, and heat flux boundary conditions and internal heat generation (see eqn. 5.15 for ODE) Forced convection, h = constant Natural convection, h = C(T-T)1/4 Radiation (eqn. 5.18) Convection and radiation (requires numerical integration) Forced convection with constant surface heat flux or internal heat generation (eqn. 5.25) (Note that the Biot number must be redefined when effects other than convection are included) 3/7/05 ME 259

Spatial Effects (Bi  0.1) LCM not valid since temperature gradient within solid is significant Need to solve heat conduction equation with applied boundary conditions 1-D transient conduction “family” of solutions: Uniform, symmetric convection applied to plane wall, long cylinder, sphere (sections 5.4-5.6) Semi-infinite solid with various BCs (section 5.7) Superposition of 1-D solutions for multidimensional conduction (section 5.8) 3/7/05 ME 259

Convection Heat Transfer: Fundamentals Reading: Incropera & DeWitt Chapter 6 3/7/05 ME 259

The Convection Problem Newton’s law of cooling (convection) “h” is often the controlling parameter in heat transfer problems involving fluids; knowing its value accurately is important h can be obtained by theoretical derivation (difficult) direct measurement (time-consuming) empirical correlation (most common) Theoretical derivation is difficult because h is dependent upon many parameters it involves solving several PDEs it usually involves turbulent fluid flow, for which no unified modeling approach exists 3/7/05 ME 259

Average Convection Coefficient Consider fluid flow over a surface: Total heat rate: If h = h(x), then 3/7/05 ME 259

The Defining Equation for h The thermal boundary layer: From Fourier’s law of conduction: Setting qconv = qcond and solving for h: 3/7/05 ME 259

Determination of Need to find T(x,y) within fluid boundary layer, then differentiate and evaluate at y = 0 T(x,y) is obtained by solving the following conservation equations: Mass (continuity) ……. Eqn (6.25) Momentum (x,y) ……. Eqn (6.26, 6.27) Thermal Energy …….. Eqn (6.28a,b) Solution of these four coupled PDEs yields the velocity components (u,v), pressure (p), and temperature (T) Exact solution is only possible for laminar flow in simple geometries 3/7/05 ME 259

Boundary Layer Approximations Because the velocity and thermal boundary layer thicknesses are typically very small, the following BL approximations apply: The BL approximations with the additional assumptions of steady-state, incompressible flow, constant properties, negligible body forces, and zero energy generation yield a simpler set of conservation equations given by eqns. (6.31-6.33). 3/7/05 ME 259

Boundary Layer Similarity Parameters BL equations are normalized by defining the following dimensionless parameters: where L is the characteristic length of the surface and V is the velocity upstream of the surface The resulting normalized BL equations (Table 6.1) produce the following unique dimensionless groups: Note: all properties are those of the fluid 3/7/05 ME 259

Boundary Layer Similarity Parameters, cont. Recall the defining equation for h: The normalized temperature gradient at the surface is defined as the Nusselt number, which provides a dimensionless measure of the convection heat transfer: 3/7/05 ME 259

The Nusselt Number The normalized BL equations indicate that Since the Nusselt number is a normalized surface temperature gradient, its functional dependence for a prescribed geometry is Recalling that the average convection coefficient results from an integration over the entire surface, the x* dependence disappears and we have: 3/7/05 ME 259

Physical Significance of the Reynolds and Prandtl Numbers Reynolds number (VL/) represents a ratio of fluid inertia forces to fluid viscous shear forces small Re: low velocity, low density, small objects, high viscosity large Re: high velocity, high density, large objects, low viscosity Prandtl number (cp/k) represents the ratio of momentum diffusion to heat diffusion in the boundary layer. small Pr: relatively large thermal boundary layer thickness (t > ) large Pr: relatively small thermal boundary layer thickness (t < ) Note that Pr is a (composite) fluid property that is commonly tabulated in property tables 3/7/05 ME 259

The Reynolds Analogy The BL equations for momentum and energy are similar mathematically and indicate analogous behavior for the transport of momentum and heat This analogy allows one to determine thermal parameters from velocity parameters and vice-versa (e.g., h-values can be found from viscous drag values) The heat-momentum analogy is applicable in BLs when dp*/dx*  0 (turbulent flow is less sensitive to this) The modified Reynolds analogy states where Cf is the skin friction coefficient 3/7/05 ME 259

Turbulence Turbulence represents an unsteady flow, characterized by random velocity, pressure, and temperature fluctuations in the fluid due to small-scale eddies Turbulence occurs when the Reynolds number reaches some critical value, determined by the particular flow geometry Turbulence is typically modeled with eddy diffusivities for mass, momentum, and heat Turbulent flow increases viscous drag, but may actually reduce form drag in some instances Turbulent flow is advantageous in the sense of providing higher h-values, leading to higher convection heat transfer rates 3/7/05 ME 259

Forced Convection Heat Transfer – External Flow Reading: Incropera & DeWitt Chapter 7 3/7/05 ME 259

The Empirical Approach Design an experiment to measure h steady-state heating transient cooling, Bi<<1 Perform experiment over a wide range of test conditions, varying: freestream velocity (u ) type of fluid (,  ) object size (L) Reduce data in terms of Reynolds, Nusselt, and Prandtl 3/7/05 ME 259

The Empirical Approach, cont. Plot reduced data, NuL vs. ReL, for each fluid (Pr): Develop an equation curve-fit to the data; a common form is typically:  < m < 1  < n <  0.01 < C < 1 3/7/05 ME 259

Flat Plate in Parallel, Laminar Flow (the Blasius Solution) Assumptions: steady, incompressible flow constant fluid properties negligible viscous dissipation zero pressure gradient (dp/dx = 0) Governing equations 3/7/05 ME 259

Blasius Solution, cont. Similarity solution, where a similarity variable () is found which transforms the PDEs to ODEs: Blasius derived: 3/7/05 ME 259

Empirical Correlations Flat Plate in parallel flow: Section 7.2 Cylinder in crossflow: Section 7.4 Sphere: Section 7.5 Flow over banks of tubes: Section 7.6 Impinging jets: Section 7.7 3/7/05 ME 259

Forced Convection Heat Transfer – Internal Flow Reading: Incropera & DeWitt Chapter 8 3/7/05 ME 259

Review of Internal (Pipe) Flow Fluid Mechanics Flow characteristics Reynolds number Laminar vs. turbulent flow Mean velocity Hydrodynamic entry region Fully-developed conditions Velocity profiles Friction factor and pressure drop 3/7/05 ME 259

Pipe Reynolds Number um is mean velocity, given by Reynolds number for circular pipes: Reynolds number determines flow condition: ReD < 2300 : laminar 2300 < ReD < 10,000 : turbulent transition ReD > 10,000 : fully turbulent 3/7/05 ME 259

Pipe Friction Factor Darcy friction factor, f, is a dimensionless parameter related to the pressure drop: For laminar flow in smooth pipes, For turbulent flow in smooth pipes, good for 3000 < ReD < 5x106 For rough pipes, use Moody diagram (Figure 8.3) or Colebrook formula (given in most fluid mechanics texts) 3/7/05 ME 259

Pressure Drop and Power For fully-developed flow, dp/dx is constant; so the pressure drop ( p) in a pipe of length L is The pump or fan power (Wp) required to overcome this pressure drop is where p is the pump or fan efficiency 3/7/05 ME 259

Thermal Characteristics of Pipe Flow Thermal entry region Thermally fully-developed conditions Mean temperature Newton’s law of cooling 3/7/05 ME 259

Newton’s Law of Cooling & Mean Temperature The absence of a fixed freestream temperature necessitates the use of a mean temperature, Tm, in Newton’s law of cooling: Tm is the average fluid temperature at a particular cross-section based upon the transport of thermal (internal) energy, Et: Unlike T , Tm is not a constant in the flow direction; it will increase in a heating situation and decrease in a cooling situation. 3/7/05 ME 259

Fully-Developed Thermal Conditions While the fluid temperature T(x,r) and mean temperature Tm(x) never reach constant values in internal flows, a dimensionless temperature difference does - and this is used to define the fully-developed condition: the following must also be true: therefore, from the defining equation for h: i.e., h = constant in the fully-developed region 3/7/05 ME 259

Pipe Surface Conditions There are two special cases of interest, that are used to approximate many real situations: 1) Constant surface temperature (Ts = const) 2) Constant surface heat flux (q”s = const) 3/7/05 ME 259

Review of Energy Balance Results For any incompressible fluid or ideal gas pipe flow, 1) For constant surface heat flux, 2) For constant surface temperature, 3) For constant ambient fluid temperature, 3/7/05 ME 259

Convection Correlations for Laminar Flow in Circular Pipes Fully-developed conditions with Pr > 0.6: Note that these correlations are independent of Reynolds number ! All properties evaluated at (Tmi+Tmo)/2 Entry region with Ts = const and thermal entry length only (i.e., Pr >> 1 or unheated starting length): combined entry length correlation given by eqn. (8.57) in text 3/7/05 ME 259

Convection Correlations for Turbulent Flow in Circular Pipes Fully-developed conditions, smooth wall, q”s = const or Ts = const Dittus-Boelter equation - fully turbulent flow only (ReD > 10,000) and 0.7 < Pr < 160 NOTE: should not be used for transitional flow or flows with large property variation 3/7/05 ME 259

Convection Correlations for Turbulent Flow in Circular Pipes, cont. Sieder-Tate equation - fully tubulent flow (ReD > 10,000),very wide range of Pr (0.7 - 16,700), large property variations Gnielinski equation - transitional and fully-turbulent flow (3000 < ReD < 5x106), wide range of Pr (0.5 - 2000) f given on a previous slide (eqn. 8.21) 3/7/05 ME 259

Convection Correlations for Turbulent Flow in Circular Pipes, cont. Entry Region Recall that the thermal entry region for turbulent flow is relatively short, I.e.,only 10 to 60D Thus, fully-developed correlations are generally valid if L/D > 60 For 20 < L/D < 60, Molki & Sparrow suggest Liquid Metals (Pr << 1) Correlations for fully-developed turbulent pipe flow are given by eqns. (8.65), (8.66) 3/7/05 ME 259

Convection Correlations for Noncircular Pipes Hydraulic Diameter, Dh where: Ac = flow cross-sectional area P = “wetted” perimeter Laminar Flows (ReDh < 2300) use special NuDh relations rectangular & triangular pipes: Table 8.1 annuli: Table 8.2, 8.3 Turbulent Flows (ReDh > 2300) use regular pipe flow correlations with ReD and NuD based upon Dh 3/7/05 ME 259

Heat Transfer Enhancement A variety of methods can be used to enhance the convection heat transfer in internal flows; this can be achieved by increasing h, and/or increasing the convection surface area Methods include surface roughening coil spring insert longitudinal fins twisted tape insert helical ribs 3/7/05 ME 259