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