Ohm's Law & Kirchoff's Law Review, & Measurement Method Concepts

1. Consider the circuit below, where current is driven by two voltage sources, one which is a known 18 V and another which is an unknown vx. Plot (on an appropriate scale) the resistance RA as a function of I1 in the range 0.001 A ≤ I1 ≤ 20 A.

2. The circuit depicted below contains a capacitor and voltage source.

(a.) If the capacitor is initially at zero charge, plot the current passing through the 30 Ω and 300 Ω resistors as functions of time.

(b.) If the voltage source is suddenly disconnected (as shown below), plot the current through the 30 Ω and 300 Ω resistors as functions of time.

3. Consider the modified bridge system depicted below (which is similar to a Wien Bridge), containing fixed resistances of magnitude R, an applied DC voltage V₁, a variable resistor RS, and a measured voltage V₂, to determine an unknown resistor and capacitor. If the capacitor is initially uncharged, in terms of the other parameters provided (including R_x and C_x), what must be the value of RS be for V₂ to be zero at all times? Show your derivation, and your answer may be a function of time, t.

4. Show how the following functions can be transformed into linear curves of the form Y = a₁X+a_o (i.e. determine Y, X, a₁ and a₀)
a. y = bx^m
b. y = be^mx
c. y = b + c*x^1/m

5. A force measurement system (i.e. a scale) has the following specifications: Range: 0 to 1000 N

Linearity Error: 0.10% Full Scale Output (FSO) Hysteresis Error: 0.10% FSO

Sensitivity Error: 0.15% FSO Zero Drift: 0.20% FSO

Estimate the overall uncertainty for this system, using the maximum possible FSO in your computations. What would the uncertainty be if the hysteresis error were removed?

6. The drag force (F_D) on a particle moving through a gas at low speed is found to be linear proportional to its speed relative to the gas(v) , i.e. F_D = -f_v. f is called the friction factor, and has units of kg s^-1. Measurements of f are carried out as a function of gas molecule mass (m_g, in kg per molecule), gas thermal energies (kT, the product of Boltzmann's constant and the temperature in Joules), mass densities, ρ_gas (kg m^-3), and dynamic viscosities (μ, in Pa s) for spherical particles of different radii, r_p. Data are provided in Table 1.

Table 1. A summary of friction factor measurements for variable radii particles under variable gas conditions.

(a.) Using Buckingham Pi type analysis, identify two or more dimensionless parameters which can be used to describe the independent variable, and the dependent variable (the friction factor).

(b.) The mean free path of a gas, A (in meters), defined as the average distance gas molecules travel between collisions with gas molecules, plays a role in determining the drag on nanoparticles. It is proportional to the ratio: (μ/ρ_gas )(m_gas/kT)^1/2, hence for non-dimensionalization this ratio can be used in lieu of the mean free path. Using only the friction factor, particle radius, viscosity, and the ratio (μ/ρ_gas )(m_gas/kT)^1/2 as variables, develop two dimensionless ratios, one to represent the independent variable, and a second for the dependent variable. Why are only two dimensionless ratios required to examine this system?

(c.) Plot, on appropriate axes, the dependent variable in (b.) as a function of the independent variable. Do all the data collapse to a single function?

7. On the following page, Table 2 displays results of heat flow experiments where two different fluids were used to cool flat plate electronic chips. The measured heat flow, equivalent to the product of the chip area, the convection coefficient, and the difference in temperature of the plate and fluid upstream temperature, was converted simply to the convection coefficient (units of W m^-2 K^-1) after experiments, and was reported as a function of the plate length (the plates were squares), fluid kinetmatic viscosity (the viscosity divided by the density), the fluid thermal diffusivity (the fluid thermal conductivity divided by the fluid density and specific heat), and the fluid thermal conductivity.

(a.) How many dimensionless ratios are needed to describe the dependent variable (the convection coefficient, h) as a function of the independent variables? Define these ratios, and also define symbols for each dimensional variable in the table. Create a table listing the values of all dimensionless parameters calculated based on the results in Table 2.

(b.) The different fluid experiments can be distinguished from one another in Table 2 by the different fluid properties provided. For each fluid, is one of the independent dimensionless ratios determined in a constant? If not, perform an alternative non-dimensionalization such that one of the independent dimensionless ratios is a constant for a given fluid.

(c.) Using the dimensionless ratios in (b.), summarize all results in a plot of the independent variable's dimensionless ratio and a single dependent variable's dimensionless ratio. You may use separate plots for different fluids. Be sure to use appropriate axes on plots.

Table 2. A summary of the measured convection coefficients for heat transfer from test electronic chips (flat plates), as functions of the plate length, upstream average velocity, working fluid kinematic viscosity, working fluid thermal diffusivity, and working fluid thermal conductivity.