Heat Transfer Today

The mathematical description of transient heat conduction yields a second-order, parabolic, partial-differential equation. When applied to regular geometries such as infinite cylinders, spheres, and planar walls of small thickness, the equation simplifies to one having a single spatial dimension. With specification of an initial condition and two boundary conditions, the mathematician uses  separation of variables to solve the equation.   That process leads to an expression for temperature distribution in the form of an infinite series. The time-honored Heisler charts use a one-term approximation to the series and present the results in graphical form.  Heat transfer practitioners have used these charts widely for the last 75 years.

This module was intended for K-12 Outreach.  However, for the more advanced user it provides plenty of opportunities to learn the fundamentals of conduction heat transfer in solids. In it we model the cooking of a roast by solving the transient heat conduction equation in a finite, axisymmetric cylindrical geometry. The solution of the governing equation takes place entirely behind the scenes. The chef can “watch” the roast heat up over time. When cooking is completed, he or she can slice the roast at positions along its length and (virtually) sample the results.  The model includes both conventional “roasting” through heat transfer at its surface and volumetric heating as occurs in a microwave oven.

The mathematical description for multi-dimensional, steady-state heat-conduction is a second-order, elliptic partial-differential equation (a Laplace or Poisson Equation). Typical heat transfer textbooks describe several methods for solving this equation for two-dimensional regions with various boundary conditions. Analytical solutions usually involve an infinite series of transcendental functions. This series is truncated and evaluated at an array of locations to give an approximate estimate of the temperatures found over the 2-D region.  Some texts also include detailed graphical methods using various paper and pen tools for estimating temperature and heat-flow lines for 2-D problems.

In this module we model the boundary layer equations for forced convection over a flat plate in their primitive form.  That is, we relax the similarity restrictions of the Blasius solution. The result  can be considered a “virtual” laboratory.   The user can take the data produced in this simulation just as in a real physical laboratory.  But in this virtual laboratory they don’t need to worry about toxicity, burning themselves, etc.   Furthermore they can work with a range of fluids that no one physical laboratory could ever hope to study.

Internal Forced Flow (Pipe Flow)

Heat transfer analysts usually handle internal (pipe) flows using convection correlations. For laminar flows the correlations are based on analysis; for transition and turbulent flows they are generally based on experiment.  Another “on-screen” laboratory, our internal flow module instead solves the fundmental energy balance equation directly.    Here we address the thermal entry length problem.  In this scenario the velocity profile is already fully developed when a change in the wall thermal boundary condition is introduced.   Our model handles laminar, transition and turbulent flows.    The thermally (and hydrodynamically) fully developed condition is, of course, the asymptotic limit of the thermal entry problem.

The user of this “laboratory” selects either a fixed wall temperature or prescribed wall heat flux.  To solve the discretized form of the governing advection-diffusion equation, our algorithm uses a single pass, space-wise marching technique.  That scheme is implicit in the radial direction and is combined with backward differencing in the axial direction.

This module covers natural convection in a fluid-saturated, permeable material.  A number of recent graduate-level heat transfer texts cover this topic. The problem is analogous to classical Rayleigh-Benard natural convection in homogeneous fluid layers. The fluid is assumed to be “Boussinesq,” i.e., the fluid density is only a function of temperature and variable only in the body force term.  In addition, it completely fill all interstices.  The Darcy equations govern the fluid motion. Heating is from the bottom of the layer or from side to side.

The normal textbook treatment of heat transfer from fins involves the solution by analytical means of a single ordinary differential equation. That equation describes transport by conduction along the fin and convection from its surface. Even with a straight, rectangular fin, there is a quandary about what thermal boundary condition to apply at the tip.  Most practitioners resolve that question using an “extended length.”  In the case of straight fins of triangular cross-section and annular fins of constant thickness, the textbooks give  solutions in terms of Bessel functions, which the student may not have studied. Also, in the latter cases users never see the temperature distribution along the fin.  Graphical representation at best shows an overall (and confusing) parameter, the fin efficiency.   The major application of extended surface heat transfer is in arrays of fins (heat sinks.)

The usual treatment of heat exchanger thermal design and analysis is based on two analysis-based solution methods applied to the governing, coupled heat balance equations.  For all but the simplest configurations, the solution of these differential equations by analytical means is challenging .   For this reason, the analytical results have been graphed in non-dimensional form.  Engineers have used the resulting charts routinely for the past three quarters of a century. The LMTD method is commonly used for heat exchanger design, that is, determining the required thermal size.   Similarly, the Effectiveness – NTU method is used for performance calculations.

Unfortunately, the charts and equations associated with these two methods do not give a complete picture of what is happening inside the exchanger, only a single overall measure. In this two-part module, our algorithm solves the same governing heat balance equations in real time in discretized form using modern numerical techniques.   The result is the same “bottom line” results as the traditional methods, but also a complete picture of what is happening within the device.  In addition, not having to look up a “correction factor” manually means the user can test parameters at will until an improved design is had.

This module solves the governing equations for natural convection next to a heated vertical flat plate in their similarity form.  We convert the continuity, vertical momentum (simplified using the Boussinesq approximation) and thermal energy equations into a pair of non-linear, coupled ordinary differential equations.  We specify the needed boundary conditions at both the surface of the plate and out at a distance from the plate.  We discretize the two equations and solve the resulting linear system iteratively.

This calculation allows the user to explore the effects of thermal boundary conditions and thermal conductivity in a simple, plane wall geometry. The user selects one of the three supplied thermal boundary conditions (fixed temperature (Dirichlet), specified heat flux (including the adiabatic condition) (Neumann) and convective boundary (Robin)), supplies numerical values of any parameters and performs the calculation. The resulting temperature distribution in the wall, computed using Fourier’s Law, is displayed.