Session-6
We will cover,
• Phase Diagrams • Enthalpy • Specific Heat Constant • Incompressible Fluid • Compressed Liquid (subcooled) • Molar Base and Universal Gas Constant • Critical State and Reduced Coordinate • Compressibility
1 Flow Chart
Determine phase of substance
Compare with saturation table
Compressed Liquid Two-phase Mixture Superheated Vapor
Direct look up Determine quality Direct look up
Interpolate other properties
2 Enthalpy
For a processes conducted at constant pressure (dP=0), balance of energy is:
dE = δQ − δW dE = δQ − pdV considering only P- V work 2 2 2 dE = dQ − p dV ∫1 ∫1 ∫1
E2 − E1 = Qp − p(V2 −V2 )
(E2 + PV2 ) − (E1 + PV1) = Qp Since E, P, and V are state variables, (E + PV ) must also be a state variable. Define Enthalpy (H ) : H ≡ E + PV
H2 − H1 = Qp
∆H = Qp The enthalpy change of the system is equal to the heat absorbed at constant pressure. Thus, ∆H is often referred to as the "heat of reaction."
H h = , h = e + Pv = u + Pv and h = (1− x) h + x h m f g 3 Specific Heat Constants energy required to raise the temperature of a system by one degree (at constant pressure or constant volume). ⎛ ∂u ⎞ cv = ⎜ ⎟ ⎝ ∂T ⎠v ⎛ ∂h ⎞ cp = ⎜ ⎟ ⎝ ∂T ⎠ p
Specific Heat Ratio
c k = p cv
When density is assumed to be constant throughout a process the process is called “incompressible” and the fluid is called “incompressible fluid.” h(T, P)= u(T )+ Pv
⎛ ∂h ⎞ ⎛ ∂u ⎞ ⎜ ⎟ = ⎜ ⎟ → cp = cv = c ⎝ ∂T ⎠ p ⎝ ∂T ⎠v
⎛ ∂u ⎞ T2 cv = ⎜ ⎟ → du = cdT → u2 − u1 = cdT ∫T ⎝ ∂T ⎠v 1
T2 h2 − h1 = cdT + P()v2 − v1 ∫T 1
5 Compressed Liquid Approximation
• If the substance is a compressed (subcooled) liquid and the compressed liquid table is unavailable or inadequate, you may invoke the compressed liquid approximation:
u(T, P) ≅ u f (T )
v(T, P) ≅ v f (T ) (weak function of T)
h(T, P) ≅ u f (T ) + Pv f (T ) = hf (T ) + []P − Psat (T ) v f (T )
• It can be interpreted as weak dependency of most properties on pressure in the compressed liquid region. • Hence, most properties can be simply approximated by their saturated liquid values at the specified temperature, except specific enthalpy.
6 Interpretation of Compressed Liquid Approximation
P-v diagram T-v diagram exact state exact state line
ressure
co Constant p nst ant tem pe ratu re line approximate state approximate state
7 Compressed Liquid Approximation
• However, the compressed liquid approximation for enthalpy is exceptional. It is due to the intrinsic definition of enthalpy being explicitly dependent on pressure: h = u + Pv • It is incorrect to neglect the pressure variation in enthalpy evaluation. Hence, h(T, P) = u(T, P) + Pv(T, P)
≅ u f (T) + Pv f (T)
= u f (T) + Psat (T)v f (T) + Pv f (T) − Psat (T)v f (T) 14244 4344 h f (T )
= hf (T) + (P − Psat (T))v f (T)
8 Molar Base and Universal Gas Constant
If molecular weight of a substance is M v = Mv v is molar specific volume
u = Mu, h = Mh, cp = Mcp
Universal Gas Constant kJ R = M.R = 8.314 kmol.K
R is called “Gas Constant” which is tabulated for different fluids.
9 Critical State and Reduced Coordinate
The critical temperature is the maximum temperature at which liquid and vapor phases can coexist in equilibrium. • Recall the phase diagrams of a general substance:
Comp. Critical Point g Liquid liquid Superheated n Critical Point e i S t n a l i t vapor e l u rat n d M io i ed at u Solid iz q v r i o l a p p a d o V e r t Two-phase li a n r e n u dome o t ti Vapor a a S im bl Su
• Base on the thermodynamic properties associated with the critical point, a non-dimensional reduced coordinate can be defined for each substance:
reduced pressure: P , T PR = reduced temperature: T = P R cr Tcr 10 Compressibility Chart
Compressibility Ideal Gas Factor: P v P v Z = = R T R T
Ideal Gas: Z =1
Good for: • low pressure critical point • high temperature
(Taken from Figure 3-56 in Cengel & Turner) 11