Page 13 - Fundamentals of The Finite Element Method for Heat and Fluid Flow
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                        INTRODUCTION
                           With respect to the laws of thermodynamics, only the first law is of interest in heat
                        transfer problems. The increase of energy in a system is equal to the difference between
                        the energy transfer by heat to the system and the energy transfer by work done on the
                        surroundings by the system, that is,
                                                      dE = dQ − dW                           (1.7)
                        where Q is the total heat entering the system and W is the work done on the surroundings.
                        Since we are interested in the rate of energy transfer in heat transfer processes, we can
                        restate the first law of thermodynamics as
                           ‘The rate of increase of the energy of the system is equal to the difference between the
                        rate at which energy enters the system and the rate at which the system does work on the
                        surroundings’, that is,
                                                     dE    dQ    dW
                                                         =    −                              (1.8)
                                                      dt   dt    dt
                        where t is the time.


                        1.4 Formulation of Heat Transfer Problems

                        In analysing a thermal system, the engineer should be able to identify the relevant heat
                        transfer processes and only then can the system behaviour be properly quantified. In this
                        section, some typical heat transfer problems are formulated by identifying appropriate heat
                        transfer mechanisms.


                        1.4.1 Heat transfer from a plate exposed to solar heat flux


                        Consider a plate of size L × B × d exposed to a solar flux of intensity q s , as shown in
                        Figure 1.1. In many solar applications such as a solar water heater, solar cooker and so
                        on, the temperature of the plate is a function of time. The plate loses heat by convection
                        and radiation to the ambient air, which is at a temperature T a . Some heat flows through
                        the plate and is convected to the bottom side. We shall apply the law of conservation of
                        energy to derive an equation, the solution of which gives the temperature distribution of
                        the plate with respect to time.
                           Heat entering the top surface of the plate:

                                                          q s A T                            (1.9)
                           Heat loss from the plate to surroundings:

                           Top surface:
                                                                   4
                                                                        4
                                               hA T (T − T a ) +  σA T (T − T )             (1.10)
                                                                        a
                           Side surface:
                                                                        4
                                                                   4
                                               hA S (T − T a ) +  σA S (T − T )             (1.11)
                                                                        a
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