EQUATIONS - THERMODYNAMICS
    

First Law - Open & Closed Systems, Processes, and Energy Balance for:
           
(Nozzles, diffusers, turbines, pumps, compressors, throttling,
             valves, boilers, condensers, evaporators, heat exchangers, 
            mixers, separators, heaters)
For detailed thermodynamic relationships for all process click here 

First Law A statement of the conservation of energy in a thermodynamic system. Net energy crossing a system boundary equals the energy change inside the system.

Heat (Q) is energy transferred due to a temperature difference and is considered positive if it is inward or added to the system.
   

Closed
Thermodynamic
System
DU = Q - W 
    DU = change in system internal energy
    Q = net heat crossing boundary
         (Q is positive if heat added to system)
    W = net work done by or on the system
         (W is positive if the system does work)
    
Closed System
 Special Cases

  
Processes For detailed P,V,T , work, Q, Enthalpy, relationships for all processes, click here
    
Constant Pressure Charles' Law
T/v = constant for an ideal gas
when pressure is constant
     W = PDV
  
Constant Volume
T/P = constant for an ideal gas
when volume is constant
     W = 0
   
Constant Temp. Doyle's Law
PV = constant for an ideal gas
when temperature is constant
   w = R'T ln(v2/v1) = R'T ln(P1/P2)
          R' = R/Mwt
   
Isentropic PVk = constant for an ideal gas
   w = (P2v2 - P1v1)/(1 - k)
   w = R'(T2 - T1)/(1 - k)
  
Polytropic PVn = constant for an ideal gas
   w = (P2v2 - P1v1)/(1 - n)
   w = R'(T2 - T1)/(1 - n)
   
Open
Thermodynamic
System
Mass can cross system boundary (flow).
Includes flow work of mass entering system.
Kinetic and Potential energies usually neglected.

Reversible work:
   wrev = xvdP + Dke + Dpe
First Law applies whether or not the processes are reversible.
  

First Law
(Energy Balance)
(In - Out) = Change
S
m'i[hi+Vi2/2a + gZi ] - Sm'e[he+Ve2/2a + gZe ]
    + Q'in - W'net = d(msus)dt

m'i = mass flow rate in.
hi = enthalpy of mass in.
Vi = Velocity of mass in.
Zi = elevation of mass in.
m'e = mass flow rate exit.
he = enthalpy of mass exit.
Ve = Velocity of mass exit.
Ze = elevation of mass exit.
Q'in= rate of heat transfer
W'net = rate of net or shaft work transfer
ms = mass of fluid with in system
us = specific internal energy of system
a = kinetic energy correction factor
     a = 1 for turbulent flow
     a = 0.5 for laminar flow
   
Special Cases of 
Open Systems
      
Constant Volume w = -v(P2 - P1)
   
Constant Pressure w = 0
  
Constant Temp Pv = constant for ideal gas
 w = R'T ln(v2/v1) = R'T ln(P1/P2)
        R' = R/Mwt
   
Isentropic PVk = constant for an ideal gas
   w = k(P2v2 - P1v1)/(1 - k)
   w = kR'(T2 - T1)/(1 - k)
Wis = [ kR'T1 / (k-1) ] [1 - (P2/P1)[(k-1)/k] ]
    
Polytropic  w = n(P2v2 - P1v1)/(1 - n)
  
Steady State Systems The state of the system does not change with time.
   
Energy Balance (In - Out) = 0
S
m'i[hi+Vi2/2a + gZi ] - Sm'e[he+Ve2/2a + gZe ]
    + Q'in - W'net = 0

Sm'i = Sm'e 
Q'  = rate of heat  transfer
W' = rate of work transfer
    
Special Cases Of Steady Flow Energy Equation
   
Nozzles/Diffusers Assumptions:
    Velocity term is significant
    No elevation change
    No heat transfer, Q = 0
    No work
    Single mass stream
hi+Vi2/2 = he+Ve2/2
efficiency = (Ve2-Vi2)/[2(hi - hes)]
    hes= enthalpy at isentropic exit state
  
Turbines, Pumps, Compressors Assumptions:
    Adiabatic - no heat transfer, Q= 0
    Velocity terms ignored.
    Work is significant
    Single mass stream
hi = he+ w

Efficiency (turbine)=  (h1 - he)/(hi - hes)
Efficiency (comp, pump)=  (hes - hi)/(he - hi)
   
Throttling Valve
Throttling Process
Assumptions:
   W = 0
   Q = 0
   Single mass stream
   Velocity terms often insignificant
hi = he
  
Boilers, Condensers, Evaporators, one side of a Heat Exchanger Assumptions:
   Heat transfer terms are significant
   Single mass stream
hi + q = he 
  
Heat Exchangers Assumptions:
   Q = 0
   W = 0
   Two mass streams (m'1, m'2)
m'1(h1i - h1e) =  m'2(h2i - h2e)
  
Mixers, Separators, Open or Closed Feedwater Heaters Smi'hi = Sm'ehe 
Smi = Sm'e

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