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THERMODYNAMICS CHAPTER 4 ENERGY ANALYSIS OF CLOSED SYSTEMS (E balance…
THERMODYNAMICS
CHAPTER 4
ENERGY ANALYSIS OF CLOSED SYSTEMS
E balance for any system
undergoing
any process
E balance in the rate form
E' in - E' out = dE syst/dt (kW)
The total quantities are related to the quantities per unit time
Q (kJ) = Q'. ∆t (etc)
Energy balance per unit mass basis
Energy balance in differential form
Energy balance for a cycle
E balance for a closed system
:red_flag:
Q net, in - W net, out = ∆E system
OR
Q - W = ∆E
Q = Q net,in = Qin - Qout
W = W net, out=W out - W in
Energy balance when sign convention is used (i.e., heat input and work
output are positive; heat output and work input are negative).
For a cycle ΔE = 0, thus Q = W.
Differential form δq-δw=de
Per unit mass 𝑞 − 𝑤 = Δ𝑒
Stationary systems 𝑄 − 𝑊 = Δ𝑈
General 𝑄 − 𝑊 = Δ𝐸
Boundary work
Work
is energy expended when a force acts through a displacement.
boundary work
For a quasi-equilibrium process (Slow process)
Constant volume
: dV = 0
So
specific heat c v
a property: related to ΔH
unit = kJ/kg · °C or kJ/kg · K.
Constant pressure
:
specific heat c p
a property: related to ΔU
unit = kJ/kg · °C or kJ/kg · K.
Constant temperature, ideal gas
:
P=(m.R.T)/V
Then
This equation is the result of applying the ideal gas assumption for the equation of state.
For real gases undergoing an isothermal (constant temperature) process, the integral in the boundary work equation would be done numerically.
Internal Energy, Enthalpy and Specific heats of Ideal gases
h = u + P.v and P.v= R.T
SO
h = u + R.T
u = u(T) and h = h(T)
du=c v (T).dT and dh = c p (T). dT
(can be applied as integration from point 1 to point 2)
Meaning
: for ideal gases, u, h, cv, and cp vary with temperature only
u2-u1=c v, avg (T2-T1)
:red_flag:
h2-h1=c p, avg (T2-T1)
for small temperature intervals:
Δu = c v. ΔT
for :red_flag: any kinds of process (with or without constant volume).
relation between c v and c p
c p = c v + R
1 more item...
ideal-gas specific heats = zero-pressure specific heats (cp0 and cv0) = The specific heats of
real gases
at low pressures
. :red_flag:
Polytropic process
PV^n = constant
constant pressure n=0
constant volume n=vo cuc
Isothermal & ideal gas: n=1
Adiabatic & ideal gas: n = k = CP/CV
k is the ratio of the specific heat at constant pressure CP to
specific heat at constant volume CV.
W = P0 (V2-V1) + 1/2 k (x2^2-x1^2)
For ideal gas:
E BALANCE =
Q transfered to the system while W from the system
H = U + P.V
Q - W other = H2 - H1
For a constant-pressure expansion
or compression process:
ΔU + Wb = ΔH
W e, in - Q out - W b = ΔU
W e, in - Q out = ΔH = m (h2-h1)
specific volume v = constant
for
INCOMPRESSIBLE SUBSTANCE
(solids, liquids)
c = c v = c p
u2-u1=c v, avg (T2-T1)
Δh = Δu + ΔP.v
For solid: ΔP.v = 0
For liquids
constant pressure process (in heaters)
Δh = ΔP.v
constant temperature process (in pumps)
Δh = Δu= c avg.ΔT
The enthalpy of a compressed
liquid
In an expansion process
W b = W friction + W atm + W crank = ∫ from 1 to 2 (F friction + Patm.A + Fcrank).ds
The area under the process curve on a P-V diagram represents the boundary work.
The
boundary work
done during a process depends on the path followed as well as the end states.
The
net work done during a cycle
is the difference between the work done by the system and the work done on the system.
OUTLINE
work
boundary work
inexact differential
quasi-equillibrium process
