Component 211: Steam to Air Heat Exchanger by HVACSIM+
General Description
This component represents a cross flow heat exchanger consisting of
horizontal tubes with circular fins. Air flowing across the finned tubes is
heated, while steam inside the tubes is condensed. The steam side is assumed
to be equipped with a trap which allows condensate, but not vapor, to leave the
tubes. No attempt is made to compute the condensate temperature at the exit.
Instead, the condensate outlet temperature is assumed to be a fixed number of
degrees below the saturation temperature. In addition, the average temperature
of steam and condensate is assumed equal to the saturation temperature, and
condensation is assumed to occur at constant pressure. The method used to
represent air temperature dynamics assumes that the dynamics are due to changes
in the steam pressure and thus in the saturation temperature, rather than to
changes in the air inlet temperature or flow rate.
Inputs to the model are the steam pressure and inlet temperature, the
air inlet temperature and flow rate, and the air outlet temperature, which is
obtained from the first output of the model. Other outputs are the condensate
outlet temperature, the mass flow rate of the team (which is determined by the
condensation rate), and the rate of heat transfer to the air.
Nomenclature
Aa - minimum air flow area (m^2)
Ai - tube inside surface area (m^2)
Ap - primary (tube) outside surface area (m^2)
As - secondary (fin) surface area (m^2)
Cm - thermal capacitance of heat exchanger (kJ/C)
Cpa - specific heat of air (kJ/kg-C)
Cpw - specific heat of liquid water (kJ/kg-C)
Df - fin diameter (m)
Di - tube inside diameter (m)
Do - tube outside diameter (m)
Fm - number of fins per meter
g - acceleration of gravity (m/s^2)
hi - inside surface heat transfer coefficient (kW/m^2-C)
ho - outside surface heat transfer coefficient (kW/m^2-C)
Hfg - latent heat of vaporization of water (kJ/kg)
Hsi - steam enthalpy at inlet (kJ/kg)
Hwo - condensate enthalpy at outlet (kJ/kg)
Ka - thermal conductivity of air (kW/m-C)
Kf - fin thermal conductivity (kW/m-C)
Kt - tube thermal conductivity (kW/m-C)
Kw - thermal conductivity of liquid water (kW/m-C)
NTU - number of transfer units ( - )
Nu - Nusselt number, based on tube outside diameter ( - )
Pr - Prandtl number ( - )
Ps - steam pressure (kPa, absolute)
Q - dynamic rate of heat transfer to air (kW)
Qss - steady state rate of heat transfer (kW)
Re - Reynolds number, based on tube outside diameter ( - )
Rf - ratio of finned tube surface area to unfinned tube surface area
Ri - inside surface heat transfer resistance (C/kW)
Ro - outside surface heat transfer resistance (C/kW)
Rr - ratio of inside surface heat transfer resistance to total
heat transfer resistance ( - )
Rt - tube heat transfer resistance (C/kW)
Tai - air inlet temperature (C)
Tao - air outlet temperature (C)
Taos - steady state air outlet temperature (C)
Tsc - difference between saturation temperature and condensate
outlet temperature (C)
Tss - saturation temperature of steam (C)
Tsi - steam inlet temperature (C)
Tt - tube inside surface temperature (C)
Two - condensed water outlet temperature (C)
wa - air mass flow rate (kg/s)
ws - steam mass flow rate (kg/s)
& - fin thickness (m)
n - fin efficiency ( - )
ua - air viscosity (kg/m-s)
uw - liquid water viscosity (kg/m-s)
pw - density of water (kg/m^3)
Toll - heat exchanger time constant (s)
Mathematical Description
The model first determines the inlet steam enthalpy, the saturation
temperature, and the outlet temperature and enthalpy, using property functions
(described in section 3.5 of reference [4]):
Hsi = HS(Ps,Tsi)
Tss = TSATS(Ps)
Two = Tss - Tsc
Hwo = HSATW(Tss) - Cpw*Tsc
Next, the outside heat transfer coefficient is computed, using a correlation
from Hausen [1] for air flow over finned tubes:
Nu = C*Re^0.625*Rf^(-0.375)*Pr^(1/3)
In this equation, the constant C is 0.30 for in-line tube rows or 0.45 for
staggered tube rows. A value of 0.30 is used in the present model. The
Prandtl number, Pr, is evaluated at the bulk air temperature, using air
property functions (described in section 3.6 of reference [4]). Rf, the ratio
of outside surface area to the area of unfinned tubes with the same outside
diameter, is calculated from the parameters. The characteristic length used in
the Reynolds number, Re, and in the Nusselt number, Nu, is the tube outside
diameter. Thus,
Re = wa*Do / (ua*Aa)
and
ho = Nu*Ka / Do
The fin efficiency, n, is determined by the use of subroutine SUFED (described
in section 3.3 of reference [4]). The outside heat transfer resistance is then
given by
Ro = [ho*(n*As + Ap)]^(-1)
The heat transfer resistance of the heat exchanger tubes is
Rt = (Do - Di) / (2*Kt*Ai)
The inside surface temperature of the pipe, Ts, and the inside heat transfer
coefficient, hi, are determined iteratively from the following equations:
hi = 0.612*[(Kw^3*pw^2*g*Hfg) / (uw*Di*(Tss - Tt))]^0.25
Tt = Tss - Rr*(Tss - Ta)
where
Rr = Ri / (Ri + Rt + Ro)
Ri = (hi*Ai)^(-1)
and the correlation for hi is from Kern [2].
The steady state rate of heat transfer is used to find the steady state air
outlet temperature and the rate of condensation, which is equal to the entering
steam mass flow rate:
Qss = hi*Ai*(Tss - Tt)
Taos = Tai + Qss/(wa*Cpa)
ws = Qss/(Hsi - Hwo)
Air temperature dynamics are computed by the approximate method of Myers et
al. [3], assuming that the thermal capacitance of the heat exchanger walls is
much greater than that of the air.
d(Tao)/dt = (Taos - Tao)/Toll
where
Toll = Cm/(Alpha*wa*Cpa)
Alpha = (2*NTU*exp(-z)*sinh(z)) /
[Rr*{(sinh(z) + cosh(z)}*exp(-z) - exp(-NTU)]
z = NTU / (2*(1 - Rr))
NTU = [wa*Cpa*(Ri + Rt + Ro)]^(-1)
Finally, the dynamic rate of heat transfer to the air is calculated:
Q = wa*Cpa*(Tao - Tai)
Component 211 Configuration
Inputs Description
1 Ps - steam pressure (kPa, absolute)
2 Tsi - inlet steam temperature (C)
3 Tai - inlet air temperature (C)
4 Tao - outlet air temperature (C)
5 wa - mass flow rate of air (kg/s)
Outputs Description
1 Tao - outlet air temperature
2 Two - outlet temperature of condensate (C)
3 ws - mass flow rate of steam (kg/s)
4 Q - rate of heat transfer to air (kW)
Parameters Description
1 Tsc - subcooling of condensate below saturation (C)
2 Aa - minimum air flow area (m^2)
3 Ai - inside heat transfer area (m^2)
4 Ap - primary (tube) outside surface area (m^2)
5 As - secondary (fin) surface area (m^2)
6 Di - tube inside diameter (m)
7 Do - tube outside diameter (m)
8 Kt - tube thermal conductivity (kJ/kg-K)
9 Df - fin diameter (m)
10 Kf - fin thermal conductivity (kJ/kg-K)
11 & - fin thickness (m)
12 Fm - number of fins per meter
13 Cm - thermal capacitance of metal (fins and tubes) (kJ/K)
Reference:
1. Hausen, H. Heat Transfer in Counterflow, Parallel Flow and Cross Flow.
New York: McGraw-Hill Book Company, 1983.
2. Kern, D.Q. Process Heat Transfer. New York: McGraw-Hill Book Company,
1950.
3. Myers, G.E., Mitchell, J.W., and Lindeman, C.F. Jr. "The transient
response of heat exchangers having an infinite capacitance rate fluid."
Transactions of the ASME: Journal of Heat Transfer, pp. 269-275,
May 1970.
4. HVACSIM+ Building Systems and Equipment Simulation Program Reference
Manual (NBSIR 84-2996)
Daniel R. Clark
United States Department of Commerce
National Institute of Standards and Technology
Gaithersburg, Maryland 20899-0001