Mathematical Modeling of Heat Through a Hot Bath

Eastern Oregon University
Eastern Oregon University
Eastern Oregon University
Talk Abstract
No one likes a cold bath. When the bath water starts to get cold a person might turn on a constant trickle of water to keep the water tepid. We attempted to model, and optimize, this behavior with differential equations. To do this we simplified the situation and started with a bathtub that had only water in it. We were able to model the heat flow through the bath water using a simple “lumped system” analysis that broke the tub up into 10 equal “plates” of water. The first of these plates was assumed to stay at a constant temperature because of the continual introduction of hot water from the tap. That first plate would then pass some of its heat to the next plate according to thermal conductance and Newton’s Law of Cooling. This model assumed that no heat was lost anywhere in the system except to the next plate of water. To add an element of realism we then added heat loss to the air above each plate. We then attempted to add a person to the tub. We treated the person as if they were a “Heat Sink”. Meaning that they take in heat but their temperature never changes. The rate of heat transfer between the water and the person required different physics from those that we had been using to model the heat flow through the homogeneous tub. We approximated this different rate by using the ratio of the thermal conductance's of water and a person. We also distributed the person’s mass evenly in each plate. This model worked reasonably well. Showing that the water in the tub would all reach the temperature of the incoming water. We then attempted to distribute the person’s mass throughout the tub in a more realistic way, with more mass towards the back of the tub. To do this we introduced a mass ratio term to our equations. This ratio of water mass to person mass did not have the desired results. The water at the back of the tub would continue to decline in temperature, seemingly unaffected by the incoming hot water. This discrepancy is mostly likely caused by the person absorbing more heat than they actually would in reality. The effect of the person’s movements was not included in our models. The shape of the tub was also not explored; our models assume either a perfectly rectangular tub or a perfectly hemispherical tub.
Talk Subject
Ordinary Differential Equations and Dynamical Systems
Time Slot
Room Number
STAG 162