Modeling the Hot Water

 

A tiny project to do some day, and an illustration of the power of modeling: analyze and simulate the temperature of the water coming out of the shower here at home.

Observations:

  • water is at first cold
  • ~30 seconds later it gets quite warm
  • another ~30 seconds later it gets somewhat cooler again
  • this phenomenon is most prominent in the winter

Hypotheses:

  • pipes running through the basement hold ~30 seconds worth of water at full flow
  • the on-demand water heater can raise the temperature of water going through it a fixed number of degrees at maximum flow rate
  • water coming into the house from outside is colder in the winter than in the summer
  • pipes themselves have heat capacity and take a while to warm up

Approach:

  • make a spreadsheet with a row for each time unit and a pair of columns for each length unit of pipe between the street and the water heater and the shower head, one cell for the temperature of the pipe, the other for the temperature of the water
  • begin with the first row at constant temperature equal to the basement temperature
  • set up formulæ to compute the next row of the spreadsheet based on the current row:

  • new extremely cold water enters on the left from the pipes in the ground outside the house * each cell recording water temperature gets the value from the cell in the row above and to its left, modified to allow for heat loss from the water to the surrounding pipe based on the temperature difference between the water and the pipe segment * each cell recording pipe temperature gets heated or cooled from its prior-row value based on the temperature difference between it and the current water temperature inside it and the basement temperature

    • Results: a spreadsheet-array of temperatures for water and pipes that exhibits behavior like what observations show. Next steps: experiment, numerically, with insulating the pipes, decreasing the flow rate, varying the water temperature coming into the house, heating the basement, etc. Fun: if you like modeling, and maybe even helpful in developing insights into what the most critical features of the situation are. Key: figuring out what are the essential factors, gathering enough data, simplifying the model so that it can be implemented, and then comparing its results with observation to see if it actually works and how accurate it is.

    (cf. TechnicalMinded (2003-07-18), Theory of Flight (2008-02-26), Pulsating Hidden World (2009-09-28), ...) - ^z - 2011-12-04