Approximate vs. Exact Solutions in Heat Transfer

In analytical heat transfer, there are a variety of techniques to determine both exact and approximate solutions to problems.  To one way of thinking, all numerical solutions are approximate, since the exact differential equations have been discretized to enable an approximate solution via a system of algebraic equations.

Both exact and approximate solutions can be useful.  Let’s look at an example, then talk about some useful features of both types of solutions.

Liquid-piston dynamometers

A dynamometer is an important tool for studying liquid piston engines (or any kind of engine, for that matter) .  A dynamometer allows us to run the engine under varying external loads and also to measure the power produced by the engine.  All kinds of dynamometers have been developed over the years suited to many different engines and purposes.  While the testing of the solar-powered engine discussed in my last post provided the proof-of-concept, and yielded some useful operational information, the fact that the engine ran completely unloaded (only operating against internal friction) imposed a pretty severe limitation on the amount and usefulness of the information that could be learned.  Essentially, it would be like evaluating a car engine if the car were never taken out of neutral.  Actually, it is even more limiting than that because at least you could rev up a car engine whereas liquid piston engines operate in a resonance, and hence single speed, mode.

A Solar-powered Fluidyne Test Bed

In our laboratory at the University of Colorado at Colorado Springs, we built and instrumented a solar-powered fluidyne (liquid piston heat engine) test bed. [1]

The test bed was intended to serve two purposes.  First, it was used to demonstrate direct solar-powered operation of a fluidyne with sunlight concentrated directly on the fluidyne cylinder rather than on a remote heat exchanger.  In addition, it was used to characterize some aspects of the operational thermodynamic cycle.  The test bed was only intended as a platform to explore feasibility and thermodynamic characteristics, not as a practical, power producing engine.  The test bed was to be powered by the sun using a 50” by 37” Fresnel  lens. The Fresnel lens supplied ample solar power for the fluidyne.  

Measurements of temperature and pressure inside the working space of the engine showed that a temperature gradient existed across the working space at all times.  

This figure shows about 20 seconds of temperature data from a single test.  During this segment, the temperature fluctuations on the hot side were about 4 °C, and the temperature fluctuations on the cold side were approximately 12 °C.  As expected, the temperature fluctuations from the two thermocouples are out of phase with one another.  

This figure shows a pressure-volume plot of the working space.  The enclosed area represents the work of each cycle.  This engine was operated without any external load for this phase of testing.  Therefore, the indicated cycle work is only overcoming internal friction and other losses.   

The instrumented solar-powered fluidyne test bed demonstrated consistent and repeatable operation.  The Fresnel lens used provided ample energy to power the fluidyne.   Indicated work of the engine was collected with pressure and volume measurements.  Since the engine was unloaded, this work only overcame frictional losses.

[1]  J.W. Mason and J.W. Stevens, 2011, “Design and Construction of a of a Solar-powered Fluidyne Test Bed” Proceedings of the ASME 2011 Mechanical Engineering Conference and Exposition IMECE2011, November 11-17, 2011, Denver, Colorado, USA, paper IMECE2011-62194.

Liquid-piston Stirling Engines

       As I indicated in my last post, one bedeviling factor in many renewable energy applications is the difficulty of making a cost-effective conversion device because it is so hard to make enough money to offset relatively high capital costs. Naturally, improvements in conversion efficiency can help on the money-making side.  Another approach would be to try to pursue technologies where the capital cost (and typically, the efficiency) is low to begin with.  If the primary energy source has a low enough cost (zero, for example) the low efficiency is not as big a concern.  Note, however, that even with free energy and low capital equipment costs, there still remain many other costs such as land, maintenance, transport, etc.  Nevertheless, technologies with low costs for capital equipment are worth exploring [1,2] and may be viable for some niche applications.

       One such technology that may be promising is the liquid piston Stirling engine.  These heat engines can be built very cheaply from a few pieces of pipe and tubing.  While they tend to have very poor efficiency and power density, their very low capital cost could make them economical for some applications using free or low cost sources of heat.

Liquid piston Stirling engines use oscillating columns of liquid, typically but not necessarily, water, in place of the pistons of a traditional Stirling engine.  This figure shows one common configuration (of many possible) with water held in a U-shaped tube and in a connecting “tuning” line.  Oscillations of water in the U-tube shift the working fluid above the water (typically air) back and forth between the hot-side (heat input) and cold-side (heat rejection) of the engine.  Oscillations of the water in the tuning column alternately compress and expand the working fluid.  If the phasing is correct, the expansion occurs when the bulk of the working fluid is on the hot side, and the compression occurs when the bulk of the working fluid is cold, and the cycle produces net work.  The work can be extracted from the oscillations of the liquid in either column, from the pressure fluctuations of the working fluid, or in other ways.  In a properly designed system, the oscillations in both the U-tube and tuning column start moving spontaneously and with the proper phasing when heat input and extraction starts with the working fluid.  An animation at the end of the post illustrates the fluid motions.

         Liquid piston Stirling engines have had limited commercial success to power water pumps and have been proposed for electricity generation in a number of configurations.  West provides a comprehensive overview of the theory and operation of these devices [3,4]. Active research continues on a variety of aspects and applications e.g. [5-7].  Because they have relatively poor power density and efficiency, they would tend to function best for small-scale applications using free or low-cost heat sources such as commercial waste heat or concentrated solar energy.  Since the power output is proportional to the mean pressure of the working fluid as well as the working fluid volume and temperature difference, atmospheric engines using water for pistons have severe limitations on improvements of the power density.  Other configurations are, of course, possible.

On the positive side, liquid piston Stirling engines can be constructed with very low capital cost as indicated previously.  In addition, they are quite robust thermally and mechanically.  They are self-starting with the input of heat, and function well within very loose design and construction tolerances. 

 References

[1] J.W. Mason and J.W. Stevens, 2011, “Design and Construction of a  Solar-powered Fluidyne  Test Bed” Proceedings of the ASME 2011 Mechanical Engineering Conference and Exposition IMECE2011, paper IMECE2011-62194.

[2] J.W. Stevens, 2010, “Low Capital Cost Renewable Energy Conversion With Liquid Piston Stirling Engines,”  Proceedings of ASME 2010 4th International Conference on Energy Sustainability ES2010, paper ES2010-90129.

[3] West, C.D., 1983, Liquid Piston Stirling Engines, Van Nostrand Reinhold, N.Y.

[4] West, C.D., 1987, Stirling Engines and Irrigation Pumping, Oak Ridge National Laboratory Technical Report ORNL/TM-10475.

[5] Orda E. and Mahkamov, K., 2004, “Development of ‘Low-tech’ Solar Thermal Water Pumps for Use in Developing Countries,” J. Sol. Energy Eng. Vol. 126, pp. 768-774.

[6] Slavin, V.S., Bakos, G.C., Finnikov, K.A., 2009, “Conversion of thermal energy into electricity via a water pump operating in Stirling engine cycle”, Applied Energy, Vol. 86, pp. 1162-1169.

[7] Van de Ven, J.D., 2009, “Mobile hydraulic power supply: Liquid piston Stirling engine pump”, Renewable Energy, Vol. 34, pp. 2317-2322.
 

Capital Costs and Renewable Energy

Sometimes people think that because the primary energy is free, use of renewable energy sources should naturally grow to fill most or all of the energy needs of the world.  However, energy from renewable sources currently constitutes a relatively small fraction of total energy use in the United States and worldwide.  As one illustration, consider the electric power generation in the U.S by primary energy source in this figure from several years ago.  Excluding hydroelectric, only about 3% of the U.S. electric power came from renewable energy.  While progress is being made, and other perspectives would give somewhat different results, the main point is that renewable sources currently contribute very little to the overall energy needs of the world.

Why is this so?  Part of the answer lies with the relatively high capital costs associated with renewable primary energy sources.

While renewable energy sources such as wind and solar eliminate fuel costs, they have a low energy density relative to hydrocarbon fuels. Consequently, collection, transformation, and transport costs tend to be much higher per unit of energy than for traditional fuels, and the overall (capital, maintenance, fuel) cost of providing the renewable-sourced energy is often not economical.  While progress in efficiency and infrastructure continues to bring costs down, there is still a long way to go before renewable technologies replace hydrocarbon fuels on a large scale.  

As an illustration, this figure shows consumer prices for small photovoltaic panels as a function of size.  A rough estimate from this admittedly unscientific survey (I just looked up prices for solar panels on the internet) indicates that PV panels can be purchased for around $7000/kW. 

Now, the next figure shows an estimate for payback period as a function of capital cost, interest rate, and selling price of electricity with the assumption of no fuel cost and neglecting maintenance cost.  For a remotely reasonable payback period in the neighborhood of five years, the capital cost would have to be in the neighborhood of $1000-$3000 per kW.  Photovoltaic panels, at least at readily available consumer prices, correspond to an unreasonably long payback (around 18 years) for even the most optimistic assumptions on electricity prices and investment costs.

 So what is the conclusion?  Simple economics dictates that many renewable energy sources currently available are priced out of a competitive (unsubsidized) market by capital cost alone, despite any attractive attributes with regard to fuel costs.  Capital cost for renewable energy conversion technologies will be a primary element in their viability.

Free Surface Profile of a Circular Hydraulic Jump

Impinging jets have excellent heat transfer characteristics and are widely used where a lot of heat has to be moved or removed rapidly.  In the case of a liquid jet directed downward against a horizontal surface, a hydraulic jump forms when the free surface makes a sudden transition from supercritical to subcritical flow.  The jump formed by the radial spread of a circular jet is sometimes called a “circular” hydraulic jump.

This work measured the hydraulic jump surface profile and unsteady fluctuations for several configurations of jet sizes, flow rates and downstream depths.  The downstream depth was controlled by an adjustable height weir shown in the figure as height “w”.

The instantaneous position of the free surface was determined by using a fine wire probe.  A small dc voltage was applied to the water, and the probe completed the circuit between the water and the power supply ground.  The fluctuating surface of the hydraulic jump made it necessary to approximate the jump profile as a statistical representation of a large number of individual measurements.

The figure below shows data for a nozzle diameter of 7.8 mm.  Each symbol represents the dimensionless depth (y/d)  where the probe was in the water 50% of the time.  The vertical error bars on each plotted point represent the 0% and 100% submersion locations.  This  figure provides a representation of the approximate mean location of the free surface, along with the approximate vertical extent of the free surface fluctuations.  As would be expected, the fluctuations are largest near the front edge of the hydraulic jump, and are greater for higher Reynolds number.  The progression from a single jump structure to a double jump structure with increasing downstream depth described by Liu and Lienhard (1993) is clearly visible.

The next figure shows the extent of the vertical fluctuations for several sets of data.  The fluctuations are largest at the front of the hydraulic jump, then drop rapidly and decrease more slowly toward the back of the jump.

 Some measurements were taken in the thin flow layer upstream of the hydraulic jump.  The figure below demonstrates a comparison of some measured layer depths with a correlation of layer depths calculated from velocity measurements in a separate work.  Overall, the comparison shows the same general trends and similar layer depths. 

 

  

References

Stevens, J.W., 1995, “Free Surface Flow Profile and Fluctuations of a Circular Hydraulic Jump Formed by an Impinging Jet,” ASME Journal of Fluids Engineering, Vol. 117, pp. 677-682.

Stevens, J., and Webb, B.W., 1992, “Measurements of the Free Surface Flow Structure Under an Impinging Free Liquid Jet,” ASME Journal of Heat Transfer, Vol. 114, pp. 79-84.

Liu, X., and Lienhard, V.J.H., 1993, “The Hydraulic Jump in Circular Jet Impingement and in Other Thin Liquid Films,” Experiments in Fluids, Vol. 15, pp. 108-116.

Transient Heat Transfer Overview

I first saw a figure like this in Professor Adrian Bejan’s book Heat Transfer  and I really liked the succinct way that it illustrates the ballpark relationship between various transient conduction heat transfer approximations that we commonly use.

In log-log coordinates, we have the Biot number (Bi), which is a dimensionless heat transfer coefficient plotted on the vertical axis, and the Fourier number (Fo), which is dimensionless time, plotted on the horizontal axis.

In a solid which is initially at a uniform temperature and is subjected to a sudden change in the boundary condition, the temperature changes will initially occur (Fo <<1) mostly very close to the surface, and the temperature field can be approximated by the solution for a semi-infinite body.  Professor Bejan calls this the “early” regime, and it is represented by blue shading on the left half of the figure.  If the heat transfer coefficient is high enough (large Bi) or for some reason the surface temperature is specified,  then the solution for a fixed surface temperature (still semi-infinite) is appropriate as shown in a different shade of blue in the upper part of the left side of the figure.

For cases where the conduction within the solid is rapid relative to the heat transfer out, the temperature variation across the body can be neglected.  The entire body can be characterized by a single temperature at any given time and the temperature variation of the body with time can be approximated with an approach called “lumped capacitance”.  The Biot number serves as a measure of when the spatial temperature variation is expected to be small relative to other temperature differences.  Lumped capacitance solutions are appropriate for Bi << 1, or a common rule of thumb that is sometimes cited is Bi < 0.1.  Professor Bejan terms this the “late” regime, and it is represented by the reddish shading in the lower right side of the figure.

For cases where neither the Fo nor the Bi number is very small, the exact solutions must be applied.  Actually, the exact solutions could be applied anywhere, but since they are in the form of infinite series, it is most convenient to use one of the simpler approximations, when appropriate.  These infinite series converge rapidly for large values of time, so a for Fo > 1, it is an excellent approximation to use only the first term of the series.  A common rule of thumb for using only the first term of these series is Fo > 0.2.  A graphical presentation of the temperature distribution based on the first term of these series has become known as the “Heisler Charts”.  The area covered by the first term of the exact solutions, or by the Heisler Charts, is shown in the upper right quadrant of the figure shaded green.

Definitions:

 L_c= a characteristic length of the solid

h = convective heat transfer coefficient at the surface of the solid

k = thermal conductivity of the solid

a = thermal diffusivity of the solid

t = time

Reference
Bejan, Adrian.  Heat Transfer.  New York:  Wiley, 1993.