Building the Model of Heat Flows

Recall from Chapter 2 that it's best to "start with the stocks" when building a flow diagram for a new model. Also, recall from Chapter 2 that that the stocks represent the storage in the system. In this example, we need two stocks:

one stock to keep track of the volume of water stored in the glass.

a second stock to represent the internal energy stored in the water.

Time will be measured in seconds; volume will be measured in cc; and any flows affecting the volume of water will be measured in cc/second. The internal energy content will be measured in calories, and any flows affecting the energy content will be measured in calories/second.

We know that water will gradually leave the glass through evaporation, so we need one flow to account for the reduction in volume as the water evaporates over time. The internal energy content will be controlled by two flows:

Energy content will be reduced as the water is evaporated. This heat flow depends on the rate of evaporation and the latent heat of evaporation.

Energy content will be increased if heat flows across the side wall of the glass. This heat flow depends on the temperature difference across the side wall, the thickness of the glass and the conductivity of the glass.

We now have two stocks and three flows. Figure 2 shows a model comprised of these stocks and flows. Then, as advised in Chapter 2, the model is completed by using converters to explain each of the flows.

Figure 2. The Water Temperature Model.

Most of the variables in Figure 2 are converters. You will probably notice that many of the variable names are shorter than names in any of the other models in the book or on the website. I have used short names for the physics exercise because shorter names will make it easier for you to check the model equations against equations in your introductory physics text.

Several of the converters in Figure 2 are inputs (like the air temperature or the density of water). Several are conversion factors. For example, three converters are used to convert Erate 1 (in feet/year) to Erate 2 (in cm/sec). Most of the units are relatively easy to identify from your introductory course in physics. But the thermal conductivity of glass requires some extra consideration. The value of "k" is 0.2 calories per second-degree C per square meter of glass. This means that the flow across 1 square meter of glass (with a thickness of 1 meter) would be 0.2 calories per second for each degree of temperature gradient across the surface.

Figure 3 shows the equations for the stocks and flows of the model. The listing also includes documentation for each variable.

Internal_Energy_Content(t) = Internal_Energy_Content(t - dt) + (HeatFlowIn - HeatLoss) * dt INIT Internal_Energy_Content = 9000 DOCUMENT: internal energy content in calories INFLOWS: HeatFlowIn = k*TempDif*SideA2/thick2 DOCUMENT: The heat flow in across the side surface in calories per sec OUTFLOWS: HeatLoss = latent_heat_of_evap*evap DOCUMENT: Heat Loss due to evaporation in calories/second Volume_of_Water_in_Glass(t) = Volume_of_Water_in_Glass(t - dt) + (- evap) * dt INIT Volume_of_Water_in_Glass = 1000 DOCUMENT: the volume of water remaining in the glass (in cubic centimeters) OUTFLOWS: evap = Erate2*TopA DOCUMENT: evaporation is measured in cubic centimeters per second

Figure 3. Equations for the stocks and flows of the water temperature model.

Figure 4 shows the equations for all of the remaining variables along with their documentation:

AirTemp = 20 DOCUMENT: the ambient air temperature in degrees C Base_Temp = 10 DOCUMENT: an arbitrary base temperature used to establish the temperature of the water circum = 2*PI*radius DOCUMENT: circumference measured around the top surface of water (cm) CM_per_FT = 30.5 DOCUMENT: conversion factor - centimeters in a foot cm_per_m = 100 DOCUMENT: conversion factor -- centimeters in a meter densityW = 1 DOCUMENT: in grams per cubic centimeter ERate1 = 2 DOCUMENT: evaporation rate in feet/year Erate2 = ERate1*CM_per_FT/(HR_per_YR*SEC_per_HR) DOCUMENT: evaporation rate in cm per second height = Volume_of_Water_in_Glass/TopA DOCUMENT: height of the water in the glass (cm) HR_per_YR = 8760 DOCUMENT: conversion factor - hours in a year int_energy_concentration = Internal_Energy_Content/massW DOCUMENT: calories of energy per gram of water; assumes an even distribution of energy k = .2 DOCUMENT: k is the conductivity of glass. It has complicated units latent_heat_of_evap = 585 DOCUMENT: 585 calories are needed to evaporate 1 gram of water massW = densityW*Volume_of_Water_in_Glass DOCUMENT: mass of the water in grams radius = 5.64 DOCUMENT: in cm SEC_per_HR = 3600 DOCUMENT: conversion factor - seconds in an hour SideA1 = circum*height DOCUMENT: area of water along the sides of the glass (square centimeters) SideA2 = SideA1/(cm_per_m*cm_per_m) DOCUMENT: side surface area measured in square meters SpecHeatW = 1.0 DOCUMENT: 1 calorie is needed to raise the temperature of 1 gram of water by 1 degree C. This is called the specific heat of water. TempDif = AirTemp-Water_Temp DOCUMENT: the difference between the air temperature and the water in degrees C thick1 = .5 DOCUMENT: thickness of the glass in cm thick2 = thick1/cm_per_m DOCUMENT: glass thickness in meters TopA = PI*radius^2 DOCUMENT: area of the top surface of water in square centimeters Water_Temp = Base_Temp+(int_energy_concentration/SpecHeatW) DOCUMENT: water temperature in degrees C

Figure 4. Equations for the remaining variables in the model.