## Process response depends on integration degree, delays & lags

Process response depends on process variable’s self-regulation

An indication of the ease with which a process may be controlled can be obtained by plotting the process reaction curve. This curve is constructed after having first stabilized the process temperature under manual control and then making a nominal change in heat input to the process, such as 10%. A temperature recorder then can be used to plot the temperature versus time curve of this change.[1]

Process response of self-regulating process variable vs. time:
vessel temperature after heating increase

Process response of integrating process variable vs. time:
vessel level after inflow increase [2]

Another phenomenon associated with a process or system is identified as the steady-state transfer-function characteristic. Since many processes are nonlinear, equal increments of heat input do not necessarily produce equal increments in temperature rise. The characteristic transfer-function curve for a process is generated by plotting temperature against heat input under constant heat input conditions. Each point on the curve represents the temperature under stabilized conditions, as opposed to the reaction curve, which represents the temperature under dynamic conditions. For most processes this will not be a straight-line, or linear, function.

Temperature gain vs. heat input of endothermic process:
heating a liquid till it boils and absorbs more heat

As the temperature increases, the slope of the tangent line to the curve has a tendency to decrease. This usually occurs because of increased losses through convection and radiation as the temperature increases. This process gain at any temperature is the slope of the transfer function at that temperature. A steep slope (high ΔT/ΔH) is a high gain; a low slope (low ΔT/ΔH) is a low gain.

Temperature gain vs. heat input of exothermic process:
heating a mixture of reactants till their reaction gives off reaction heat;
melting a plastic till its flow gives off frictional heat

This curve follows the endothermic curve up to the temperature level D. At this point the process has the ability to begin generating some heat of its own. The slope of the curve from this point on increases rapidly and may even reverse if the process has the ability to generate more heat than it loses. This is a negative gain since the slope ΔT/ΔH is negative. This situation would actually require a negative heat input, or cooling action.

Process response depends on process variable’s dead time delays & capacity lags

Temperature response vs. time of single capacity:
vessel-temperature lag

Temperature response vs. time of single capacity with dead time:
vessel-temperature lag with hot-water piping delay

Temperature response vs. time of two capacities:
vessel-temperature lag and vessel-wall lag

Temperature response vs. time of three capacities:
vessel-temperature lag, vessel-wall lag, and thermowell-wall lag

Two characteristics of these curves affect the process controllability, (1) the time interval before the temperature reaches the maximum rate of change, A, and (2) the slope of the maximum rate of change of the temperature after the change in heat input has occurred, B. The process controllability decreases as the product of A and B increases. Such increases in the product AB appear as an increasingly pronounced S-shaped curve on the graph.

The time interval A is caused by dead time, which is defined as the time between changes in heat input and the measurement of a perceptible temperature increase. The dead time includes two components, (1) propagation delay (material flow velocity delay) and (2) exponential lag (process thermal time constants).

Temperature responses vs. time of temperatures I, II, III, and IV

The maximum rate of temperature rise is shown by the dashed lines which are tangent to the curves.

The tangents become progressively steeper from I to IV. The time interval before the temperature reaches the maximum rate of rise also becomes progressively greater from I to IV.

As the S curve becomes steeper, the controllability of the process becomes increasingly more difficult. As the product of the two values of time interval A and maximum rate B increases, the process controllability goes from easy (I) to very difficult (IV). Response curve IV, the most difficult process to control, has the most pronounced S shape.[1]

1. Stevenson, John. “Control Principles.” Process/industrial instruments and controls handbook, 5th ed., edited by Gregory K. McMillan, McGraw-Hill, 1999, pp. 2.4-2.30.
2. Shinskey, F. Greg. “Fundamentals of Process Dynamics and Control.” Perry’s chemical engineers’ handbook, 8th ed., edited by Don W. Green, McGraw-Hill, 2008, pp. 8-5 – 8-19.

## Sticky prices are responses to customer friction

Sticky prices illustrated [1]

Sticky prices delay price jumps hoping customers will be happier

“We can’t change prices biannually, it is not the culture here.”

“We said we weren’t going to raise prices that year and I believe that once you say that, you should stick with it.”

…if the costs are stable, then doubling the frequency of price changes… invites customers to complain, to demand discounts and rebates, and to ask to renegotiate.

…price rigidity was perceived by the company’s customers to be a sign of “customer orientation” and therefore a good thing.

…many customers were more positively disposed to do business with companies who only changed their prices according to a predictable time schedule. Indeed, price rigidity was a source of pride within a company because it indicated that one’s relationship with customers was more important than the “bottom line.”

“We will take it in the pants rather than pass it on down to our customers.”

Sticky prices delay price drops that might jump up again

One member of the sales force aptly described cutting prices as “feeding the animal.” Such a decision sets up a dangerous cycle: cutting prices in order to get business this period leads to a response by a competitor with a still lower price. This lower price puts return pressure on the firm to lower its prices again.

…both the sales force and customers would sometimes argue against a price decrease because it would make a price increase in later years more expensive because of the need to convince customers that prices should go up again. Thus any price change that does not make sense for the customer can cause customer antagonism.

Sticky prices eventually jump, and effort with customers jumps up

…“every time you have one of those price changes you have to go in there and you are opening a Pandora’s box.”

“It is getting to be a running joke that every December and January I am coming in with some [price] change… They will say things like: ‘Where does that come from?… The direction is not consistent… You change discounts… dramatically, we don’t know if you are committed to us or not.’”

“Pricing season around here lasts longer than the NFL.”

“All of these costs depend on the size of the price change.”

During the pricing season we studied, a major customer called a senior vice president to negotiate a new discount level. The senior vice president and his staff flew to meet with the customer, which took two days. The team then returned to headquarters to gather additional data about the customer, similar customers, the firm’s competitors, and the effect of the customer’s purchases on the firm’s revenue. The pricing team recalculated the effect of their price changes on that customer and similar customers. They met, suggested additional analysis, met again, and decided on what they wanted to offer at the next round of meetings with the customer. Then they planned a presentation for the customer. The team then flew back with three corporate people, an area manager, and the account manager for another two days.

New large accounts require even more effort.

Although the company carries only about 8,000 products and it changes the list prices of almost all of them each year, the actual number of price changes it undertakes each year is many times higher because of the individually negotiated prices, discounts, and rebates. Therefore, the actual number of price changes undertaken is quite large, in the range of 10,000–54,000 each year.

Sticky prices speed up when overall prices move more, because people adapt

“There was… a period of some rapid inflation back in the Carter years where we would barely get a price sheet printed and you would have to start working on another one, every 6 months or so.”

“The [price] increases we experienced during that [inflationary] time were very much largely driven by cost and our average costs were going up and we were trying to recoup that… [During] high-inflation period you could get away with the high price increases. I think there was expectations in the market place; our customers are saying ‘I am able to in ate my prices to the end user so I shouldn’t be surprised when my vendor raises their prices…’ The distributors could pass on their prices a lot of easier than they can now.”[2]

1. Koning, JP. “Are prices getting less sticky?” 14 Oct. 2015, jpkoning.blogspot.com/search?q=are+prices+getting+less+sticky Accessed 13 May 2017.
2. Zbaracki, Mark J., et al. “Managerial and customer costs of price adjustment: direct evidence from industrial markets.” Review of Economics and statistics2 (2004): 514-533s.

## Control difficulty comes from less capacity and more dead time

Figure 3.9. Temperature measurement device located downstream.

Control difficulty is increased by dead time

The dead time can be computed as the length divided by the speed.

…in Figure 3.9. As the mixing of steam and cold water to produce hot water occurs in the tank, the temperature of the water in the tank is the true indicator of the state of the process. When the hot water temperature transmitter is located downstream, the temperature controller does not know the current value of the temperature of the water in the tank. Instead, the controller only knows what the temperature was at some time in the past. In essence, the controller is taking actions based on old information, which is never a good idea.

You do not know where you are; you only know where you have been. Your driving performance is seriously impaired. Furthermore, if you insist on driving this way, take some advice: slow down!

Actually, we have to give this same advice to the temperature controller for the hot water process. We do this by reducing the controller gain.

When the dead time is large, the gain must be reduced substantially, resulting in a very slow loop. The usual complaint in large dead time processes is that the controller is so slow that it is useless.

As the dead time increases, more overshoot is generally observed, and… oscillations will become pronounced.

The larger the dead time,… the slower the response.[1]

Control difficulty is reduced by capacity

Capacity is where a process stores variable amounts of mass or energy.[2]

The time constant… is always the ratio of holdup to throughput. For material balances, the holdup is material and the throughput is material flow. For energy balances, the holdup is energy and the throughput is energy flow.

Control difficulty can be quantified

…controls… can cope with time constants… far better than with dead time.[1]

The parameter

…has the property 0 ≤ control difficulty ≤ 1…

…processes with small control difficulty are easy to control, and the difficulty in controlling the system increases as control difficulty increases. Systems with control difficulty = 1 correspond to processes with pure time delay, which are difficult to control well.[3]

…dead time is the “difficult element to control.”[1]

1. Smith, Cecil L. Practical process control: tuning and troubleshooting. John Wiley & Sons, 2009, pp. 72, 83, 85, 87-90.
2. Shinskey, F. Greg. Process control systems: application, design and tuning. 4th ed., McGraw-Hill, Inc., 1996, p. 22.
3. Åström, Karl Johan, and Tore Hägglund. Advanced PID control. ISA-The Instrumentation, Systems and Automation Society, 2006, p. 26.