Component 613: Hot Water to Air Heating Coil (Detailed) by HVACSIM+
General Description
Like Component 612, this subroutine represents a water-to-air
cross-flow heating coil. Unlike Component 612, a constant overall heat
transfer coefficient is not assumed: separate inside and outside heat transfer
coefficients are calculated as functions of the water and air flow rates. In
addition, a more complicated method is used to represent temperature dynamics.
The differential equations used to calculate temperature dynamics in this model
are derived from transfer functions for the responses of the air and water
outlet temperatures to perturbations in the water inlet temperature.
Nomenclature
Ca - capacitance rate (specific heat times mass flow rate) of air
Cm - thermal capacitance (specific heat times mass) of coil material
Cmin - minimum of Ca and Cw
Cw - capacitance rate (specific heat times mass flow rate) of water
Ka - flow resistance parameter on air side of coil
Kw - flow resistance parameter on water side of coil
NTU - number of transfer units
R - ratio of minimum to maximum capacitance rate
s - Laplace operator
UA - overall heat transfer coefficient
wa - mass flow rate of air
ww - mass flow rate of water
e - effectiveness
n - fin efficiency
Toll - coil time constant
Tollc - capacitive term of coil time constant
Tollx - coil flush time
Subscripts
a - air
i - inlet
o - outlet
ss - steady state
w - water
Mathematical Description
Separate inside and outside heat transfer coefficients are calculated
as functions of the water and air mass flow rates, ww and wa:
(n*h*A)a = a*wa^b
(h*A)w = c*ww^d
The parameters a,b,c, and d may be determined from experimental data or
calculated from existing correlations (e.g. reference [1], pp. 341 and 351),
which are generally given in terms of the Reynolds number. The overall heat
transfer coefficient is then given by
UA = [(n*h*A)a^(-1) + (h*A)w^(-1)]^(-1)
and the NTU (number of transfer units) value is, by definition,
NTU = UA/Cmin
If a constant UA is desired, b and d may be set to zero.
The coil model uses and approximate equation for effectiveness, e, as a
function of NTU, for a cross-flow heat exchanger with both fluids unmixed [2]:
e = 1 - exp{[exp(-R*n*NTU) - 1]/(R*n)}
where
n = NTU^(-0.22)
The steady state air and water outlet temperatures are found using the
definition of e:
Taoss = Tai + (Twi - Tai)*e*Cmin/Ca
Twoss = Twi - (Taoss - Tai)*Ca/Cw
Coil time constants are also calculated by the model, using an
assumption that all dynamics are due to changes in the water inlet temperature.
Gartner and Harrison [3] give the following approximate transfer function for
the water outlet temperature response to perturbations in the water inlet
temperature:
g1 = T'wo(s) / T'wi(s) ~=
[exp(-gamma) + exp(-beta1)*Tollw*s]*exp(-Tollx*s) / (1 + Tollw*s)
where
Tollw = Tollx / alpha * exp(beta1*beta3 / (2*alpha))
alpha = beta3 + 2*beta2 / (2 + beta4)
gamma = beta1 - (beta1*beta3)/alpha
beta1 = (h*A)w / Cw
beta2 = (n*h*A)a*Tollx / Cm
beta3 = (h*A)w*Tollx / Cm
beta4 = (n*h*A)a / Ca
and n, Ca, Cw, and Cm are defined in the nomenclature section. T'wo and T'wi
represent perturbations from some base temperature. Using the air inlet
temperature as the base temperature, they are replaced by (Two - Tai) and
(Twi - Tai), respectively. Treating Tai as a constant, and replacing the
steady state solution with the steady state temperature calculated from the
coil effectiveness, the following equation can be derived by inverse
transformation of the transfer function:
d(Two)/dt = (1/Tollw)*{Tai - Two +
DELAY[Tollw*exp(-beta1)*(d(Twi)/dt) + Twoss - Tai]}
where DELAY is the transport delay function (described in section 3.1 of
reference [5]). Under some conditions, the accuracy and reliability of the
model may be limited by the characteristics of DELAY. For further discussion
of the limitations, see section 4.2 of reference [5].
A similar procedure is applied to the air side of the coil. Bhargava
et al. [4] give an approximate transfer function of the following form:
g2 = T'ao(s) / T'wi(s) = a1 / (1 + a2*Tollx*s + a3*Tollx^2*s^2)
Reference [4] gives closed-form expressions for the constants a1, a2, and a3,
which are derived from a more exact transfer function by the Pade polynomial
method. This approximate transfer function can be inverse transformed into a
second order differential equation, which can be separated into a pair of
coupled first order equations. Assuming a constant air inlet temperature as
before, the steady-state air temperature can be substituted into the equations.
The resulting pair of first-order differential equations is
d(Td)/dt = (Taoss - Tao) / Tollx
d(Tao)/dt = (Td - a2*Tao) / (a3*Tolla)
where
a2 = (alpha + beta1) / (alpha*gamma) - gamma'*e1
a3 = a2^2 - [(alpha + beta1)^2 - alpha*gamma]/(alpha^2*gamma^2) +
e1*[gamma'*(gamma'/2 + gamma'/gamma + 2/alpha) - 1/alpha]
gamma' = 1 + (beta1*beta3 / alpha^2)
e1 = exp(-gamma) / (1 - exp(-gamma))
and the parameters alpha, gamma, and the beta parameters are defined above. Td
is a dummy variable, introduced in the process of separating the second order
equation into two first order equations.
Model Verification
Component 613 heating coil model was tested by using experimentally
measured temperatures and flow rates as inputs to the model, and comparing the
measured air and water outlet temperatures with the calculated values. In all
cases, the agreement between simulated and measure outlet temperatures is quite
good. Similar results are obtained for step changes in the air inlet
temperature and the air flow rate. At least some of the differences must be
attributed to experimental error, since the rate of energy transfer calculated
from the measure water flow rate and steady state temperatures is not quite
equal to the energy transfer rate calculated from the measured air flow rate
and steady state temperatures.
Component 613 Configuration
Inputs Description
1 Pao - outlet air pressure
2 wa - air mass flow rate
3 Tai - inlet air temperature
4 Tao - outlet air temperature (from first output)
5 Pwo - outlet water pressure
6 ww - water mass flow rate
7 Twi - inlet water temperature
8 Two - outlet water temperature (from second output)
9 Td - dummy temperature used to calculate air
temperature dynamics (from third output)
Outputs Description
1 Tao - outlet air temperature
2 Two - outlet water temperature
3 Td - dummy temperature used internally to calculate
air temperature dynamics
4 Pai - inlet air pressure
5 Pwi - inlet water pressure
Parameters Description
1 a - coefficient for the heat transfer
correlation (n*h*A)a = a*wa^b
2 b - coefficient for the heat transfer
correlation (n*h*A)a = a*wa^b
c - coefficient for the heat transfer
correlation (h*A)w = c*ww^d
4 d - coefficient for the heat transfer
correlation (h*A)w = c*ww^d
5 Ka - flow resistance parameter, air side
6 Kw - flow resistance parameter, water side
7 Vol - water side volume of coil
8 Cm - mass times specific heat of coil material
Reference:
1. Chapman, Alan J. Heat Transfer, 3rd edition. New York: Macmillan
Publishing Co., Inc. (1974).
2. McQuiston, F.C., and Parker, J.D. Heating, Ventilating, and Air
Conditioning Analysis and Design, 2nd edition. New York: John Wiley
and Sons, Inc. (1982).
3. Gartner, J.R., and Harrison, H.L. "Dynamic characteristics of
water-to-air cross-flow heat exchangers." ASHRAE Transactions Vol. 71,
part 1 (1965).
4. Bhargava, S.C., McQuiston, F.C., and Zirkle, L.D. "Transfer functions
for crossflow multirow heat exchangers." ASHRAE Transactions Vol. 81,
part 2, pp. 294-314 (1975).
5. 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