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

Steady-State Conduction Heat Transfer

Incropera & DeWitt coverage:

–Chapter 2: General Concepts of Heat Conduction

–Chapter 3: One-Dimensional, Steady-State Conduction

–Chapter 4: Two-Dimensional, Steady-State Conduction

General Concepts of Heat Conduction

Reading: Incropera & DeWitt

Chapter 2

Generalized Heat Conduction
Fourier’s law, 1-D form:
Fourier’s law, general form:
-q” is the heat flux vector, which has three components; in Cartesian coordinates:
(magnitude)
The Temperature Gradient
ÑT is the temperature gradient, which is:
–a vector quantity that points in direction of maximum temperature increase
–always perpendicular to constant temperature surfaces, or isotherms
(Cartesian)
(Cylindrical)
(Spherical)
Thermal Conductivity
k is the thermal conductivity of the material undergoing conduction, which is a tensor quantity in the most general case:
–most materials are homogeneous, isotropic, and their structure is time-independent; hence:
which is a scalar and usually assumed to be a constant if evaluated at the average temperature of the material
Total Heat Rate
Total heat rate (q) is found by integrating the heat flux over the appropriate area:
k and Ñ T must be known in order to calculate q” from Fourier’s law
–k is usually obtained from material property tables
–to find ÑT, another equation is required; this additional equation is derived by applying the conservation of energy principle to a differential control volume undergoing conduction heat transfer; this yields the general Heat Diffusion (Conduction) Equation
Heat Diffusion (Conduction) Equation
For a homogeneous, isotropic solid material undergoing heat conduction:
Cylindrical and spherical coordinate system forms given in text (p. 64-65)
This is a second-order, partial differential equation (PDE); its solution yields the temperature field, T(x,y,z,t), within a given solid material
Heat Diffusion (Conduction) Equation
For constant thermal conductivity (k):
For k = constant, steady-state conditions, and no internal heat generation
–this is known as Laplace’s equation, which appears in other branches of engineering science (e.g., fluids, electrostatics, and solid mechanics)
Boundary Conditions and Initial Condition
Boundary Conditions: known conditions at solution domain boundaries
Initial Condition: known condition at t = 0
Number of boundary conditions required to solve the heat diffusion equation is equal to the number of spatial dimensions multiplied by two
There is only one initial condition, which takes the form
–where Ti may be a constant or a function of x,y, and z
Types of Boundary Conditions for Conduction Problems
Specified surface temperature, e.g.,
Specified surface heat flux, e.g.,
Specified convection (h, T¥ given), e.g.,
Specified radiation (e, Tsur given), e.g.,
Solving the Heat Diffusion Equation
Choose a coordinate system that best fits the problem geometry.
Identify the independent variables (x,y,z,t), e,g, is it a S-S problem? Is conduction 1-D, 2-D, or 3-D? Justify assumptions.
Determine if k can be treated as constant and if
Write the general heat conduction equation using the chosen coordinates.
Reduce equation to simplest form based upon assumptions.
Write boundary conditions and initial condition (if applicable).
Obtain a general solution for T(x,y,z,t) by some method; if impossible, resort to numerical methods.
Solving the Heat Diffusion Equation, cont.
Solve for the constants in the general solution by applying the boundary conditions and initial condition to obtain a particular solution.
Check solution for correctness (e.g., at boundaries or limits such as x = 0, t = 0, t ® ¥ , etc.)
Calculate heat flux or total heat rate using Fourier’s law, if required.
Optional: rearrange solution into a nondimensional form
Example:
GIVEN: Rectangular copper bar of dimensions L x W x H is insulated on the bottom and initially at Ti throughout . Suddenly, the ends are subjected and maintained at temperatures T1 and T2 , respectively, and the other three sides are exposed to forced convection with known h, T¥.
FIND: Governing heat equation, BCs, and initial condition
One-Dimensional, Steady-State Heat Conduction
Reading: Incropera & DeWitt,

Chapter 3

1-D, S-S Conduction in Simple Geometries w/o Heat Generation
Plane Wall
–if k = constant, general heat diffusion equation reduces to
–separating variables and integrating yields
–where T(x) is the general solution; C1 and C2 are integration constants that are determined from boundary conditions
1-D, S-S Conduction in Simple Geometries w/o Heat Generation
Plane Wall, cont.
–suppose the boundary conditions are
–integration constants are then found to be
–the particular solution for the temperature distribution in the plane wall is now
1-D, S-S Conduction in Simple Geometries w/o Heat Generation
Plane wall, cont.
–The conduction heat rate is found from Fourier’s law:
–If k were not constant, e.g., k = k(T), the analysis would yield
»note that the temperature distribution would be nonlinear, in general
1-D, S-S Conduction in Simple Geometries w/o Heat Generation
Electric Circuit Analogy
–heat rate in plane wall can be written as
–in electrical circuits we have Ohm’s law:
–analogy:
Thermal Circuits for Plane Walls
Series Systems
Parallel Systems
Thermal Circuits for Plane Walls, cont.
Complex Systems
Thermal Resistances for Other Geometries Due to Conduction
Cylindrical Wall
Spherical Wall
Convective & Radiative Thermal Resistance
Convection
Radiation
Critical Radius Concept
Since the surface areas of cylinders and spheres increase with r, there exist competing heat transfer effects with the addition of insulation under convective boundary conditions (see Example 3.4)
A critical radius (rcr) exists for radial systems, where:
–adding insulation up to this radius will increase heat transfer
– adding insulation beyond this radius will decrease heat transfer
For cylindrical systems, rcr = kins/h
For spherical systems, rcr = 2kins/h
Thermal Contact Resistance
Thermal contact resistance exists at solid-solid interfaces due to surface roughness, creating gaps of air or other material:
Thermal Contact Resistance
R”t,c is usually experimentally measured and depends upon
–thermal conductivity of solids A and B
–surface finish & cleanliness
–contact pressure
–gap material
–temperature at contact plane
See Tables 3.1, 3.2 for typical values
EXAMPLE
Given: two, 1cm thick plates of milled, cold-rolled steel, 3.18mm roughness, clean, in air under 1 MPa contact pressure
Find: Thermal circuit and compare thermal resistances
1-D, S-S Conduction in Simple Geometries with Heat Generation
Thermal energy can be generated within a material due to conversion from some other energy form:
–Electrical
–Nuclear
–Chemical
Governing heat diffusion equation if k = constant:
S-S Heat Transfer from Extended Surfaces (i.e., fins)
Consider plane wall exposed to convection where Ts>T¥:
How could you enhance q ?
–increase h
–decrease T¥
–increase As (attach fins)
Fin Nomenclature
x = longitudinal direction of fin
L = fin length (base to tip)
Lc = fin length corrected for tip area
W = fin width (parallel to base)
t = fin thickness at base
Af = fin surface area exposed to fluid
Ac = fin cross-sectional area, normal to heat flow
Ap = fin (side) profile area
P = fin perimeter that encompasses Ac
D = pin fin diameter
Tb = temperature at base of fin
1-D Conduction Model for Thin Fins
If L >> t and k/L >> h, then the temperature gradient in the longitudinal direction (x) is much greater than that in the transverse direction (y); therefore
Another way of viewing fin heat transfer is to imagine 1-D conduction with a negative heat generation rate along its length due to convection
Fin Performance
Fin Effectiveness
Fin Efficiency
–for a straight fin of uniform cross-section:
–where Lc = L + t / 2 (corrected fin length)
Calculating Single Fin Heat Rate from Fin Efficiency
Calculate corrected fin length, Lc
Calculate profile area, Ap
Evaluate parameter

Determine fin efficiency hf from Figure 3.18, 3.19, or Table 3.5
Calculate maximum heat transfer rate from fin:
Calculate actual heat rate:
Maximum Heat Rate for Fins of Given Volume
Analysis:
“Optimal” design results:
Fin Thermal Resistance
Fin heat rate:
Define fin thermal resistance:
Single fin thermal circuit:
Analysis of Fin Arrays
Total heat transfer =
heat transfer from N fins +
heat transfer from exposed base
Thermal circuit:
–where
Analysis of Fin Arrays, cont.
Overall thermal resistance:
Example
Given: Annular array of 10 aluminum fins, spaced 4mm apart C-C, with inner and outer radii of 1.35 and 2.6 cm, and thickness of 1 mm. Temperature difference between base and ambient air is 180°C with a convection coefficient of 125 W/m2-K. Contact resistance of 2.75×10-4 m2-K/W exists at base.
Find: a) Total heat rate w/o and with fins
b) Effect of R”t,c on heat rate
Two-Dimensional, Steady-State Heat Conduction
Reading: Incropera & DeWitt

Chapter 4

Governing Equation
Heat Diffusion Equation reduces to:
Solving the HDE for 2-D, S-S heat conduction by exact analysis is impossible for all but the most simple geometries with simple boundary conditions.
Solution Methods
Analytical Methods
–Separation of variables (see section 4.2)
–Laplace transform
–Similarity technique
–Conformal mapping
Graphical Methods
–Plot isotherms & heat flux lines
Numerical Methods
–Finite-difference method (FDM)
–Finite-element method (FEM)
Conduction Shape Factor
The heat rate in some 2-D geometries that contain two isothermal boundaries (T1, T2) with k = constant can be expressed as
–where S = conduction shape factor
(see Table 4.1)
Define 2-D thermal resistance:
Conduction Shape Factor, cont.
Practical applications:
–Heat loss from underground spherical tanks: Case 1
–Heat loss from underground pipes and cables: Case 2, Case 4
–Heat loss from an edge or corner of an object: Case 8, Case 9
–Heat loss from electronic components mounted on a thick substrate: Case 10

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