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