Component 505: Conduit (Duct or Pipe) by HVACSIM+
General Description
The conduit model is designed to account for three effects: thermal
losses to ambient conditions, transport delays, and dynamics due to thermal
capacitance. The model has four modes, determined by the last parameter. If
the absolute value of MODE is 1, air properties are used and the model
represents a duct. If the absolute value of MODE is 2, water properties are
used and the model represents a pipe. The sign of MODE determines the method
used to calculate thermal capacitance effects. The merits of the two methods
are discussed below.
Nomenclature
A - surface area of conduit
Cm - thermal capacitance (mass * specific heat) of conduit
Cp - specific heat of fluid
h - heat transfer coefficient
K - flow resistance coefficient
P - pressure
T - temperature
t - time
U - overall heat transfer coefficient
V - volume of conduit
w - mass flow rate of fluid
p - density of fluid
Toll - time constant
Tollc - capacitance term of time constant
Tollx - transport time
Subscripts
amb - ambient
i - inlet or inside
o - outlet or outside
ss - steady state
Mathematical Description
The pressure at the inlet of the conduit is calculated from the outlet
pressure and the mass flow rate, using a flow resistance coefficient, K, which
is assumed constant:
Pi = Po + sign(w) * K * w^2 (1)
The steady state outlet temperature is given by
Tss = Tamb + (Ti - Tamb)*exp(-gamma) (2)
where
gamma = UA/(w*Cp)
If MODE is positive, the temperature dynamics are represented by the following
equation:
d(To)/dt = (Tss - To)/Toll (3)
where
Toll = [hi/(hi + ho)]*(Cm/(w*Cp))
A derivation of this equation can be found in reference [1]. This equation is
solved analytically within the subroutine, bypassing the MODSIM, the main
HVACSIM+ program, non-linear differential equation solver:
To+ = Tss + (To- - Tss)*exp(-DeltaT / Toll) (4)
This result is then sent to the DELAY subroutine, which returns the current
outlet temperature:
To = DELAY(To+,Tollx) (5)
where Tollx is the transport time through the conduit:
Tollx = pV/w
If MODE is negative, the temperature dynamics are represented by the following
equation:
d(To)/dt = [-exp(-alpha)/Tollc]*
[To - DELAY{Tss + Tollc*exp(-gamma - alpha)*d(Ti)/dt}] (6)
where
Tollc = Cm/(hi*A)
alpha = gamma*hi/(2*ho)
and gamma is defined above. This equation is derived by inverse transformation
of an approximate transfer function given in reference [2], and is solved
outside the subroutine by the MODSIM non-linear differential equation solver.
Model Verification
The approximate transfer function from which equation 6 is derived has
been compared with a more exact transfer function [2]. The error of the
approximation is a function of alpha and Tollc*w, where w is an oscillation
frequency, but the maximum error is a function of alpha alone. When alpha is
less than or equal to one, the error never exceeds 10% in magnitude and 8
degrees in phase angle. (As discussed below, some additional error will be
introduced by the transport delay subroutine.) For further details see
reference [2]. Since alpha is directly proportional to the length of the
conduit, accuracy can be improved by dividing a long conduit into two or more
shorter conduits.
For a unit step increase in the inlet temperature, equation 6 yields an
outlet temperature step increase of magnitude exp(-hi*A/(w*Cp)) at the end of
the delay time, followed by an exponential rise with a time constant of
Tollc*exp(-alpha). The time constant for the simpler model, Toll, is
essentially always smaller than this, which helps to compensate for the lack of
an output step change in the simpler model.
Limitations of the Model
Although a step change in the inlet temperature has been applied here
for purposes of illustration, use of the detailed mode with changes in the
inlet conditions, which are rapid relative to the transport time, is not
generally recommended. This is due to the intrinsic limitations of the
transport delay subroutine, which is required to pass the derivative of the
inlet temperature in the detailed mode. A step change in the inlet temperature
produces a spike in the term sent to DELAY, and a ramp input produces a step
change in DELAY. The detailed mode is generally unreliable for step change
inputs, and its accuracy diminishes when the flow rate is such that the
transport time exceeds the duration of a ramp input or the period of a
sinusoidal input. The characteristics of DELAY also limit the accuracy of the
simple mode when the transport time exceeds the time interval between two step
changes or the period of a sinusoidal input.
Component 505 Configuration
Inputs Description
1 w - fluid mass flow rate
2 Po - outlet pressure
3 Ti - inlet temperature
4 Tamb - ambient temperature
5 To - outlet temperature
Outputs Description
1 To - outlet temperature
2 Pi - inlet pressure
Parameters Description
1 hi*A - inside surface heat transfer coefficient * area
2 ho*A - outside surface heat transfer coefficient * area
3 Cm - thermal capacitance of conduit material
4 V - volume of conduit
5 K - flow resistance coefficient
6 H - height of outlet above inlet
7 MODE - -2 => water, detailed dynamics (one D.E.)
-1 => air, detailed dynamics (one D.E.)
1 => air, simple dynamics (no D.E.)
2 => water, simple dynamics (no D.E.)
References:
1. Clark, D.R., Hill, C.R., and Hurley, C.W., "Dynamic models for HVAC
system components." ASHRAE Transactions, Vol. 91 (1985).
2. Tobias, J.R., "Simplified transfer function for temperature response of
fluids flowing through coils, pipes or ducts." ASHRAE Transactions,
vol. 79, pp. 19-22, 1973.
3. 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