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

FM, HTX, MTX, AERO, COMBUSTION

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.


Analogy Between Fluid Flow, HTX and MTX

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.

Analogy - MTX / HTX

1D Heat Transfer with Convective Boundaries and Heat Generation

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

Convective BC with Heat Generation

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!

Convective BC with Heat Generation
Finally, the temperature distribution function is expressed as:
Convective BC with Heat Generation

Plug Flow between Parallel Plates

Plug Flow

Flow in an Annular Passage

Annular Duct
Annular Duct

Flow thrugh Pipe Branches forming Tee or Wye Network

Tee - Pipe Network

Note velocities in pipe branches 1-2, 2-3 and 3-4 are not known in advance and hence the loss coefficients KMN cannot be adjusted to same velocity. Note that the junction losses at node 2 can be incorporated either in K12 or in K23 and K24. It is more convenient to club the node loss in K12 as appropriate assignment into K23 and K24 will involve extra calculations.

Tee - Pipe Network

Denoting the pressures in terms of head of water column:
Tee - Pipe Network

Simplifying further:
Tee - Pipe Network

This is a non-linear equation in V23 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.


Tee - Pipe Network

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


Equation of State: Soave-Redlick-Kwong Equation

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 m3 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 CO2 and Pitzer acentric factor are: Tc = 304.2 K, pc = 72.9 atm, and ω = 0.225.

General cubic equation of state is given by following equation.
EOS SRK
ROS SRK
Alternatively, the equation can be solved using Newton's method as described below.
ROS SRK
Iterate till the difference in vnew and vold falls to desired accuracy.
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