1. 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

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

3. 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

4. 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

5. 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

6. 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

7. General LCM
LCM can be applied to systems with convection, radiation, and heat flux boundary conditions and internal heat generation (see eqn 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

8. 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 )
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

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

10. 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

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

12. 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

13. 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

14. 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. ( ).
3/7/05
ME 259

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

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

22. 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

23. 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

24. 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

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

26. 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

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

28. 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

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

30. 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 < 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

31. 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

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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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

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

40. 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

41. 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

42. 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