Question 21.35: Transport lag is a term used to describe the time delay that......
Transport lag is a term used to describe the time delay that may be present in certain engineering systems. A typical example would be a conveyor belt feeding a furnace with coal supplied by a hopper (see Figure 21.13). The amount of fuel supplied to the furnace can be varied by varying the opening at the base of the hopper but there is a lime delay before this changed quantity of fuel reaches the furnace. The time delay depends on the speed and length of the conveyor. Mathematically, the function describing the variation in the quantity of fuel entering the furnace is a time-shifted version of the function describing the variation in the quantity of fuel placed on the conveyor (see Figure 21.14). Let
u(t)q(t) = quantity of fuel placed on the conveyor,
where u(t) is the unit step function,
u(t — d)q(t — d) = quantity of fuel entering the furnace,
d = time delay introduced by the conveyor,
l = length of conveyor (m),
v = speed of conveyor (m s^{-1}).
The input to the conveyor u(t) q(t). The output from the conveyor is u(t – d) q(t – d). If the conveyor is moving at a constant speed then v=\frac{l}{d} \text { and so } d=\frac{l}{v} \text {. } The transfer function that models the conveyor belt can be obtained by using the second shift theorem:
Q_{\mathrm{d}}(s)=\mathcal{L}\{u(t-d) q(t-d)\}=\mathrm{e}^{-s d} \mathcal{L}\{q(t)\}=\mathrm{e}^{-s d} Q(s)The transfer function for the conveyor is shown in Figure 21.15. Transport lags can cause difficulty when trying to control a system because of the delay between taking a control action and its effect being felt. In this example, increasing the quantity of fuel on the conveyor does not lead to an immediate increase in fuel entering the furnace. The difficulty of controlling the furnace temperature increases as the transport lag introduced by the conveyor increases.
