Re: A Heat Transfer Textbook Lienhard Solution 14
Beatris Ninh
Between them, the authors have spent more than 100 years using various textbook solutions manuals. In our experience, none are without errors. If you happen to find one in our solutions manual, do let us know.
Figure 18.3:Geometry of heattransfer finA model configuration is shown in Figure 18.3. The fin isof length . The other parameters of the problem are indicated.The fluid has velocity and temperature . Weassume (using the Reynolds analogy or other approach) that the heattransfer coefficient for the fin is known and has the value . Theend of the fin can have a different heat transfer coefficient, whichwe can call .The approach taken will be quasi-one-dimensional, in that thetemperature in the fin will be assumed to be a function of only.This may seem a drastic simplification, and it needs someexplanation. With a fin cross-section equal to and a perimeter, the characteristic dimension in the transverse direction is (For a circular fin, for example, ). The regimeof interest will be taken to be that for which the Biot number ismuch less than unity, , which is arealistic approximation in practice.The physical content of this approximation can be seen from thefollowing. Heat transfer per unit area out of the fin to the fluidis roughly of magnitude per unit area. Theheat transfer per unit area within the fin in the transversedirection is (again in the same approximate terms)
Figure 18.4:Element offin showing heat transferIf there is little variation in temperature across thefin, an appropriate model is to say that the temperature within thefin is a function of only, , and use aquasi-one-dimensional approach. To do this, consider an element,, of the fin as shown in Figure 18.4. There isheat flow of magnitude at the left-hand sideand heat flow out of magnitude at the right hand side. There is also heat transfer aroundthe perimeter on the top, bottom, and sides of the fin. From aquasi-one-dimensional point of view, this is a situation similar tothat with internal heat sources, but here, for a cooling fin, ineach elemental slice of thickness there is essentially a heatsink of magnitude , where is the areafor heat transfer to the fluid.The heat balance for the element in Figure 18.4can be written in terms of the heat flux using ,for a fin of constant area:
The activities of our group are in the broad area of heat and mass transfer, thermodynamics, and fluid dynamics. Our focus is on technologies for desalination of seawater and brackish water, remediation of various waste waters, and recycling of water, with increased energy efficiency and reduced environmental impact as core objectives. This work includes thermodynamic cycle analysis, transport processes in components, solar-energy driven systems, and both thermal and membrane separations.
In a previous paper [1], I studied the relationship between two-dimensional (2D) heat conduction inside and outside closed curves, showing that the conduction shape factor had the same value for the exterior region and the interior region. The primary method of that study was conformal mapping, showing that the total heat flow between the isothermal sections of the boundary was invariant under mapping. In this study, I address the local heat flux distributions on the inside and outside of the boundary, which can differ widely even though the total heat transfer on either side is the same. I develop a boundary integral formulation of the interior/exterior conduction problem, using methods from classical potential theory, and I apply that formulation to identify the relationship between interior and exterior heat flux, as well as their connection to the shape factors.
We have considered heat conduction inside and outside 2D closed curves that have two isothermal segments at different temperatures separated by two adiabatic segments. We have formulated the mixed boundary value problem using simple and double layer potentials, under the condition of zero net heat transfer to the far field. The results found are:
The design of a small, inexpensive temperature controlled bath (0.25 ml volume) for electrophysiological studies of isolated cells is described. The design creates a uniform temperature (1 C) from ambient temperature to approximately 37 C over a bath chamber area of 0.61 cm for flow rates from zero to 2ml/min. Access for microelectrodes and unobstructed preparation viewing with an inverted microscope is afforded by a finely stranded heater sandwiched between two glass cover slips and located bencath the bath chamber. Battery operation of the feedback temperature controller and solution pre-heater minimizes electrical interference.