Analyze the Data
Students will analyze and graph the data taken.
Learning Objectives
Students will
explore rates of change and accumulation in context
Materials
Computer and Internet connection
Calculating
Concentrations :
24° C
0%
Measured
Concentration
0° C
100%
To calculate a rate of flow, the concentration of ice water at each instant must be determined. For now, assume that the room temperature of the water was around 24° C. Two known concentration levels are: (1) when there is no ice water and (2) when there is 100% ice water. Recall that ice water is used since the temperature is known to be approximately 0° C.
Think about the following diagram.What do you think the measured temperature will be when the concentration is 50% ice water (half ice water and half room temperature water)?
How could you use the measured temperature to determine the concentration of ice water mixing with room temperature water?
Write an equation which expresses the concentration as a function of the measured temperature, the "room" or ambient temperature of the water, and the temperature of the
injectate.
The work above depends upon the two items being the same type of fluid. Notice that the concentration can be calculated without knowing the amount of ice water or the amount of room temperature water. In the case of ice water and blood additional factors must be taken into account. For our simplified model, the relationship above is sufficient. Click here if you wish to learn more about the physics of heat and the calculation of the temperature of the mixture.
the concentration of ice water at each data point.
Calculating Flow Rates:
Quantity = Flow x Concentration x Duration
In this experiment, the total quantity of ice water is known and we are trying to determine the flow rate. Solving for the flow, we get that:
Quantity
Flow
=
Concentration x Duration
or that
30
ml
Flow
=
0.5 x Duration
At any given instant, the concentration can be determined by the temperature of the mixture. However, the concentration is not constant. Imagine that the graph of temperature over time was:
In this case, the total amount of ice water being pumped can be determined by calculating and summing the amount of ice water being pumped on each interval.
Quantity = Sum of (Flow x Concentration x Duration) of each interval Since the flow rate of the water is assumed to be a constant, the flow rate is the same for each interval, so it can be factored out of the sum, and this equation can be rewritten as:
Quantity = Flow x ( Sum of Concentration x Duration of each interval) Solving for the flow rate:
Quantity
Flow
=
Sum of Concentration x Duration
of each interval
Graph your (time, concentration) and (temperature, concentration) data. Explain the behavior of these graphs in terms of this situation.
Use your expression for the concentration and the last equation for the flow rate to calculate an approximate flow rate for your data.
How does your approximation compare to the measured flow rate ?
Summarize how the flow rate of the pumped water could be determined without knowing the total amount of water being pumped.
Data Analysis & Probability 9-12 Understand histograms, parallel box plots, and scatterplots and use them to display data.
For univariate measurement data, be able to display the distribution, describe its shape, and select and calculate summary statistics.
Compute basic statistics and understand the distinction between a statistic and a parameter.
R.A. Rhoades and G. A. Tanner, Medical Physiology, Little, Brown and Company, 1995, passage pp. 271-2, diagram pages 231 and 373. Ware, Wendy (1999). Fluoroscope of Cardiac Output being measured in a canine, Iowa State University, College of Veterinary Medicine, Ames, Iowa. Cornette, J., Ackerman, R., Keller, B. and G. Johnston (2000), Calculus for
the Life Sciences , draft manuscript, to be published by Prentice Hall.
