Optimization done from 0 to 3000 minutes with curve
weighting to make errors in fit approximately for
Temperature Drop and Mass Evaporation.
// MODEL NUMBER: 0313
/* MODEL NAME:Botijo
SHORT DESCRIPTION: Water cooling by evaporation from an earthenware jug.
Model fits data sets Temp and Mass. Analog to lung exchange.
*/
import nsrunit; unit conversion on;
unit kcal = 1000 calorie;
math Botijo {
// INDEPENDENT VARIABLE
realDomain t min; t.min=0; t.max=4800.0; t.delta=1;
// DEPENDENT VARIABLES
real
Tl(t) K, // Temperature of bulk water
V(t) cm^3, // Volume of water
hw(t) cm, // Drop in water height
As(t) m^2, // Interior water surface area
Se(t) m^2, // Wet exterior surface area
AT(t) m^2; // Total surface area
// PARAMETERS
real
hc kcal/(hr*m^2*K), // Exterior heat transfer coefficient
H = 0.011 scalar, // Enthalpy of H2O in 39 C air
Hs = 0.018 scalar, // Enthalpy of H2O in saturated 39 C air
k = 80 kg/(hr*m^2), // Mass transfer coefficient of H2O
Cp = 1.0 kcal/(kg*K), // Heat capacity of H2O
shw = 0.24 kcal/(kg*K), // Wet heat of air
Le = 1.15 scalar, // Lewis number
Tg = 312.15 K, // Air temperature of atmosphere
Ts = 297.352 K, // Surface temperature of water/jug
R = 10 cm, // Radius of Botijo
LH2O = 583 kcal/kg, // Latent heat of vaporization at 24.2
U = 22 kcal/(hr*m^2*K), // Overall heat transfer coefficient
rho = 1 g/cm^3, // Density of water
fepi = 0.622 scalar, // Radiative heat transfer coefficient
sig = 4.9E-8 kcal/(hr*m^2*K^4); // Stefan-Boltzmann constant
// INITIAL CONDITIONS
when (t=t.min) {
Tl = 312.15;
hw=6.5;
}
// EQUATIONS -------------------------
V=4/3*PI*R^3-(PI/3)*(3*R*hw^2-hw^3);
As=PI*(2*R*hw-hw^2);
Se=2*PI*R*(2*R-hw);
AT=As+Se;
hc = shw*Le^(2/3)*k;
real fepisig =fepi*sig;
// ORDINARY DIFFERENTIAL EQUATIONS
hw:t=k/rho*AT*(Hs-H)/(PI*hw*(2*R-hw));
V:t=-k/rho*AT*(Hs-H);
Tl:t=1/(V*rho*Cp)*(hc*AT*(Tg-Ts)+fepi*sig*(4*PI*R^2-Se)*
(Tg^4-Ts^4)-U*AT*(Tl-Ts)-LH2O*k*AT*(Hs-H));
} // END
/*
DETAILED DESCRIPTION:
Model is from the paper by J.I. Zubizarreta and G. Pinto (see referernces)
The model exhibits the basis of evaporative heat loss, as in the panting
of dogs or the normal water and heat loss occuring with normal breathing
due to evaporation from tracheal walls.
This model provides a "quantitative explanation of an ancient method
of chilling water." Earthenware pitchers with a spout and handel
(botijo in Spanish) have been used to hold and chill drinking water.
The slightly porous, clay-based pot allows water to transfer
through the wall of the pitcher and evaporate into the ambient
environment. The energy needed for evaporation (i.e., latent heat
of vaporization) is balanced by a reduction in the internal
energy of the liquid inside the botijo.
REFERENCES:
Zubizarreta, J.I. and Pinto, G. An Ancient method for cooling
water explained by mass and heat transfer. Chem. Eng. Educ. 1995, 29, 96.
KEY WORDS:
Mass, heat, transfer, evaporation, cooling, botijo, quantitative, education,
chilling, water, ceramic, porous
REVISION HISTORY:
Written 01/28/2011 Joseph C. Anderson
Description amended by JBB 14mar11.
JSim SOFTWARE COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
JSim software was developed with support from NIH grants HL088516,
and HL073598. Please cite these grants in any publication for which
this software is used and send one reprint of published abstracts or
articles to the address given below. Academic use is unrestricted.
Software may be copied so long as this copyright notice is included.
Copyright (C) 1999-2009 University of Washington.
Contact Information:
The National Simulation Resource,
Director J. B. Bassingthwaighte,
Department of Bioengineering,
University of Washington, Seattle, WA
98195-5061
*/
0.0 40.5 120.1 169.9 219.6 279.4 359.0 438.6
1593.1 1632.9 4310.2
0.0 45.4 128.65 174.9 230.4 276.65 359.9 433.91
1488.44 1516.19 3190.49
0.0 11.4 20.9 30.4 58.8 77.8 106.3 115.8
125.2 163.2 182.2 210.6 277.0 343.4 428.8 1595.5
1624.0 4308.4
0.08 1.97 3.78 5.2 8.03 9.37 10.79 11.69
12.2 13.19 13.5 13.82 14.33 14.61 14.84 13.78
13.78 0.35