EXAM I Name:________________

SPRING 2003

BAE 3023

Linearization:

1. The Herschel-Bulkely Equation describing Shear stress, t, for non-newtonian fluids has been modeled as:

Where:

t _o = t _yield = shear stress at inception of flow

k = consistency index

n = flow behaviour index

= shear rate in the fluid

Determine a linear form of the equation relating shear stress to shear rate at a shear rate of .

let:

2. The following transfer function was proposed to approximate the change of density of grain to changes in humidity during aeration of the grain. The time units of the equation are in hours.

[kg/⁰C]

If this equation were correct and the effects of a step change in humidity were examined,

a) How long after the humidity change would there start to be some change in density? _2.1 hours__________________________

b) How long after the temperature change would it take until the density stabilized (95% of its change)? 3*25+2.1 =77.1 hours_____

c) Would the density increase or decrease for increasing humidity? Decrease

d) If the grain were exposed to a sinusoidally varying humidity of 10% with a frequency of 1/24 cycle per hour for a long period of time, what would the magnitude of the density variation be?

e) If the grain were exposed to a sinusoidally varying humidity of 10% with a frequency of 1/24 cycle per hour for a long period of time, what would the frequency of the density variation be? __1/24 cycle per hour_______

3. A step test was performed to determine the response characteristics of a temperature measuring device. The following chart resulted as a response to a 6 C⁰ step in temperature initiated at 10 sec from the start of the chart. Estimate a transfer function that would characterize the output voltage to temperature response of the device.

4. Develop a transfer function that relates the flow q_out leaving the reservoir in the following diagram to the evaporation from the surface of the reservoir, e. Assume the outflow has units of m³/s and evaporation has units of m³/s-m² = m/s. Assume that the flow out of the reservoir can be estimated with an orifice equation as .

Assume vertical walls on the pond as per problem discussion

inflow rate – outflow rate = accumulation rate

linearizing the outflow equation using previous results:

for steady state:

subtracting the steady state equation and writing in terms of deviation variables:

Substituting the linearized outflow equation

Transforming and collecting terms:

Writing in terms of a transfer function: