Given are p 1 = 311 kN / m 2 , T 1 = 333 K , T 2 = 853 KFind p 2 ratio V 1 / V 2 and R if ω = 372 , 32 kJ/kg

Question

Given are       p    1    =  311   kN      /        m    2    ,         T    1    =  333     K  ,         T    2    =  853     KFind       p    2   ratio       V    1        /        V    2   and R if   ω  =  372  ,  32   kJ/kg

Shaniya Yang · Accepted Answer

We start with the work done in the reversible adiabatic (isentropic) process, given by:                    (1)                    w        =                              R            (                          T              1                        –                          T              2                        )                                1            −            k                              Since the initial and final temperatures are given, we have two unknowns, R (specific gas constant, which you need to determine) and kFor an ideal gas,                    (2)                    k        =                              c            p                                c            v                              where       c    v   , the specific heat at constant volume, can be calculated as follows.From first law:  Δ  u  =  q  –  wwith   q  =  0 (adiabatic process) and   Δ  u  =  –  w, where   w is positive if work is done by the gas (expansion) and negative if work is done on the gas (compression). Since the adiabatic processes increases the temperature (      T    2    &amp;gt;      T    1  ) the process must be compression, thus   w is negative.  w  =  −  372.32  kJ/kg  Δ  u  =  −  (  −  372.32  kJ/kg  )  =  +  372.32  kJ/kgBut also, for an ideal gas in a closed system,   Δ  u depends only on the temperature change and is given by the following that applies to any process (not just constant volume)  Δ  u  =      c    v    Δ  T  =      c    v    (      T    2    –      T    1    )  ,then                    (3)                              c          v                =                              +            372.32                                              T              2                        –                          T              1                                          Next we introduce the relationship between specific heats and the specific gas constant for an ideal gas:      c    p    –      c    v    =  Rthus                    (4)                              c          p                =                  c          v                +        R            recall (2)   k  =      c    p        /        c    v   and rewrite                    (5)                              c          p                =        k                  c          v                    From (4) and (5)  k      c    v    =      c    v    +  R                    (6)                    k        =        (                  c          v                +        R        )                  /                          c          v                    substitute eq (3) for       c    v   in eq (6), we then have k in terms of R, which can be substituted for k in equation (1) to solve for R. After solving R putting the value of  into equation (1) can solve for kThat leaves       v    1        /        v    2   and       p    2   to be determined. We get these from the equation for a reversible adiabatic process, and the general gas equation for an ideal gas.Process equation (reversible adiabatic- isentropic):  P      v    k    =  c  o  n  s  t  a  n  t      P    1        v    1    k    =      P    2        v    2    k                      (7)                                          v            1                                v            2                          =                              (                                          P                2                                            P                1                                      )                                1            k                              General gas equation:                    P        1                    v        1                    T      1        =                    P        2                    v        2                    T      2                          (8)                                          v            1                                v            2                          =                                            P              2                                      T              1                                                          P              1                                      T              2                                          Since k has been determined, we have two equations ( (7) and (8) ) and two unknowns      v    1        /        v    2   and       p    2   which can then be solved.

Given are p_1=311 kN/m^2, T_1=333 K, T_2=853 Asked are p_2, ratio V_1/V_2, and R if w=372,32 kJ/kg

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