JNTU B.Tech III PROCESS CONTROL INSTRUMENTATION Previous Question Paper (November/December - 2020 - R15)

A thermometer can be modeled by considering the heat balance between the process fluid and the sensing element. Let T_i be the temperature of the surrounding fluid, T_o be the temperature indicated by the thermometer, m be the mass of the sensing element, C be the specific heat capacity, A be the surface area, and U be the overall heat transfer coefficient.

The rate of heat transfer to the thermometer is given by the energy balance equation:

Rearranging this equation gives:

Dividing by UA, we obtain:

Defining the time constant τ = mC/UA, the equation becomes:

Taking the Laplace transform of both sides, assuming zero initial conditions:

The transfer function G(s) is defined as the ratio of the output to the input:

Since the transfer function is in the form 1 / (τs + 1), which is the standard form for a first-order system, the thermometer is proven to be a first-order system. The response of the thermometer to a step change in temperature follows an exponential curve characterized by the time constant τ, which represents the time required for the thermometer to reach approximately 63.2% of the total change in temperature.

For a first-order system subjected to a ramp input, the input temperature change is given by T_b(t) = A*t, where A = 10 units/min. The transfer function for a first-order thermometer is G(s) = 1 / (τs + 1), where τ = 0.2 min. The response of the thermometer to a ramp input is given by the equation:

The difference between the bath temperature and the indicated temperature, known as the dynamic error or lag, is defined as:

Given the values A = 10, τ = 0.2, and t = 1.0 min, we substitute these into the error equation:

Calculating the exponential term, e^(-5) is approximately 0.00674. Therefore:

The difference between the indicated temperature and the bath temperature after 1.0 minute is 1.9865 units.