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Textbook Solutions: Fluid Mechanics, Heat & Mass Transfer, Aerodynamics

This page is being continously updated with complex (text-book type) problems in Fluid Mechanics, Heat Transfer, Aerodynamics, Mass Transfer, Combustion and Thermodynamics. The solutions to compressible flows including sub-sonic, sonic and supersonic flows inside a converging-diverging nozzle will be presented.

The method of relating two phenomena with some come feature is a powerful tool to both understand and remember the concepts. Following table summarizes key parameters which are analogous in the field of fluid flow (momentum transfer), heat transfer and mass transfer.

The designs of heat exchangers are the most predominant application of conductive heat transfer with such boundary conditions. The temperature profile in walls of a water-cooled internal combustion engine is also subject to this combination of boundary conditions though heat generation is not present in this application. Some applications can be current carrying conductor cooled by convection on both inner and outer diameters, nuclear fission in a annular cross-section cooled by convection on both the inner and outer radii.

The governing equation and general solution of the differential equation is given by

Heat flow is assumed positive from left to right. Hence, the convective heat transfer on left face is negative if ambient temperature is < wall temperature on the left face!

Finally, the temperature distribution function is expressed as:

Note velocities in pipe branches 1-2, 2-3 and 3-4 are not known in advance and hence the loss coefficients K_{MN} cannot be adjusted to same velocity. Note that the junction losses at node 2 can be incorporated either in K12 or in K_{23} and K_{24}. It is more convenient to club the node loss in K_{12} as appropriate assignment into K_{23} and K_{24} will involve extra calculations.

Denoting the pressures in terms of head of water column:

Simplifying further:

This is a non-linear equation in V_{23} and needs to be solved using trial-and-error or iterative approaches. Microsoft Excel Goal Seek utility can be used to solve this equation as well. A good initial guess would be required.

A more compact approach in terms of fluid resistances can be used as demonstrated below.

Thus, we have 3 equations and 3 unknowns – but they are still non-linear and needs to be solved using iterative method.

For gases at very high pressure and those in liquid state such as Liquified Petrolium Gas (LPG), the accuracy of ideal gas equation falls sharply. Many improvizations have been made and SRK equation is one such method. The approach is demonstrated using following sample problem:

A gas cylinder with a volume of 5.0 m^{3} contains 44 kg of carbon dioxide at T = 273 K. estimate the gas pressure in [atm] using the SRK equation of state. Critical properties of CO_{2} and Pitzer acentric factor are: Tc = 304.2 K, pc = 72.9 atm, and ω = 0.225.

Iterate till the difference in v

The content on CFDyna.com is being constantly refined and improvised with on-the-job experience, testing, and training. Examples might be simplified to improve insight into the physics and basic understanding. Linked pages, articles, references, and examples are constantly reviewed to reduce errors, but we cannot warrant full correctness of all content.

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