Identifying of Unsteady Performance of Oil to Water Heat Exchanger Integrated with Process

2.1 Heat exchanger system

The heat exchanger process system was hypothetically separated into two parts, namely, the heat exchanger part and an abstract oil process part comprised of all other than the heat exchanger subprocesses. The proposed heat exchanger system included a hot oil stream coming from a thermal unit process which is cooled by a cold-water circuit. This system is equipped with the measuring devices and other required accessories as illustrated in Figure 1. The operating conditions of the heat exchanger is illustrated in the Table 1.

1.png

Figure 1. Schematic of heat exchanger integrated with a process

Table 1. Operating conditions of the heat exchanger

2.2 Heat exchanger state-space model

There are three common approaches to modeling the heat exchanger depending on how much information is available to start with, namely, White, Black and Grey box models. White box models require detailed knowledge regarding heat exchanger specifications which may not always be available. On the other hand, black-box models offer simplicity and need only input-output data but at the expense of flexibility. Grey-box models exploit most of the above two models since they can be developed based on structures of white-box models like the state-space structure without prior knowledge of detailed specification. A single state variable Grey-box model was suggested for the oil part. The oil mass inventory is the oil confined in the process volume and results in some latency effects [24]:

$m_{o i l} C_{p o} \frac{d x_{1}}{d t}=Q+\dot{m}_{o} C_{p o} T_{i}-\dot{m}_{o} C_{p o} x_{1}$            (1)

or as a state-space model with two inputs.

$A=\left[-\frac{\dot{m}_{o}}{m_{o i l}}\right], \quad B=\left[\begin{array}{ll}\frac{\dot{m}_{o}}{m_{o i l}} & \frac{1}{m_{o i l} C_{p o}}\end{array}\right]$          (2)

Figure 2 shows the output of this model with T_i taken from best fit to the measured oil inlet temperature. This left the model with two unknowns, i.e., oil mass inventory and external heat input, to be adjusted for the closest fit with outlet temperature. However, the theory couldn’t fit the measurement flawlessly, possible causes are either random noise or unaccounted for states.

2.png

Figure 2. Process oil input-output relationship

Figure 3 presents the error autocorrelation in which all correlations but at zero lag, are within the confidence bounds indicating probable random noise [26]. Thus, a need for a better and maybe higher-order model is recommended following the outcome of cross-correlation which indicates a possible correlation between the error and the input.

There are many methods by which the heat exchanger can be modeled depending on how much information is available to start with. In the current case, there are only measured temperatures at outlets and inlets without any specification for the process or the heat exchanger. Suitable parameters including oil and water inventories, heat addition, and conductance divisor factor (f) are chosen to fully identify the heat exchanger from measured temperatures and flow rates. Many published works are available on such cases ranging from artificial neural network (ANN) [27] to system identification [28] and other fitting methods [29]. The state-space method was selected in the current work with some unknown parameters left to be determined from measured data.

3.png

Figure 3. Error auto and cross correlations

The heat exchanger could be divided into 3 separate subsystems, water, oil, and solid material of the wall, assuming negligible axial conduction. Each one has its state, hence:

$C_{w} \frac{d T_{w}}{d x}=\dot{m}_{w} C_{p w} T_{w i}-\dot{m}_{w} C_{p w} T_{w e}-f U A\left(T_{w}-T_{S}\right)$              (3)

$C_{s} \frac{d T_{s}}{d x}=f U A\left(T_{w}-T_{s}\right)+(1-f) U A\left(T_{o}-T_{s}\right)$           (4)

$C_{o} \frac{d T_{o}}{d x}=\dot{m}_{o} C_{p o} T_{o i}-\dot{m}_{o} C_{p o} T_{o e}-(1-f) U A\left(T_{o}-T_{S}\right)$           (5)

where, T_w, T_o, and T_s are the bulk temperatures of the water, oil, and solid masses within the heat exchanger respectively. C_w, C_o, and C_s are the corresponding heat capacitances, implicitly encompass water and oil inventory masses and solid material mass within the heat exchanger. With no other information available, a linear profile is assumed for temperature variation throughout the heat exchanger. These equations were used to find the equilibrium values and then rearranged into a linearized space state model as follows:

$\left[\begin{array}{l}\dot{x}_{1} \\ \dot{x}_{2} \\ \dot{x}_{3} \\ \dot{x}_{4}\end{array}\right]$$=\left[\begin{array}{cccc}-a_{11} & 0 & 0 & a_{11} \\ 0 & a_{22} & a_{23} & 0 \\ 0 & a_{32} & a_{33} & a_{34} \\ a_{41} & 0 & a_{43} & a_{44}\end{array}\right]\left[\begin{array}{l}x_{1} \\ x_{2} \\ x_{3} \\ x_{4}\end{array}\right]$$+\left[\begin{array}{cccc}0 & \frac{1}{m_{o} C_{p o}} & 0 & 2 \frac{\left(-x_{1}^{e}+x_{4}^{e}\right)}{m_{o}} \\ 2 \frac{\dot{m}_{w}{ }^{e} C_{p w}}{C_{w}} & 0 & 2 \frac{C_{p w}}{C_{w}}\left(-x_{2}^{e}+T_{w i}^{e}\right) & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 \frac{C_{p o}}{C_{o}}\left(x_{1}^{e}-x_{4}^{e}\right)\end{array}\right]\left[\begin{array}{c}T_{w i} \\ Q \\ \dot{m}_{w} \\ \dot{m}_{o}\end{array}\right]$              (6)    

where:

x1: bulk oil temperature perturbation within the process.

x2: bulk water temperature perturbation within the heat exchanger.

x3: bulk solid temperature perturbation within the heat exchanger.

x4: bulk oil temperature perturbation within the heat exchanger.

$\mathrm{a}_{11}=2 \frac{\dot{m}_{o}}{m_{o i l}}$

$\mathrm{a}_{22}=-\frac{1}{C_{w}}\left(2 \dot{m}_{w} C_{p w}+f U A\right)$

$\mathrm{a}_{23}=\frac{f U A}{C_{w}}$

$\mathrm{a}_{32}=\frac{f U A}{C_{S}}$

$\mathrm{a}_{33}=-\frac{U A}{C_{S}}$

$\mathrm{a}_{34}=\frac{(1-f) U A}{C_{S}}$

$\mathrm{a}_{41}=\frac{2 \dot{m}_{o} C_{p o}}{C_{o}}$

$\mathbf{a}_{43}=\frac{(1-f) U A}{C_{o}}$

$\mathrm{a}_{44}=-\frac{1}{C_{o}}\left(2 \dot{m}_{o} C_{p o}+(1-f) U A\right)$

The superscript "e" denotes equilibrium conditions. From the steady measurements of the heat exchanger, UA was estimated to be around 0.2 kW/K. This UA should be divided between oil and water sides, by which (1-f) UA goes to the oil side and f UA to the water side. The value of conductance divisor factor (f) is between 0 and 1 and adjusted according to experimental data.

The fluctuation in the process heat is to be alleviated by controlling oil and water flow rates. The figure of merit here is the bulk oil temperature (x₁) since it is an indication of the temperature in the process itself.

Although, T_wi is being handled with the input matrix, but it is more like a disturbance than a controlled input. The cooling water supply is a very large reservoir and assumed of constant temperature, however; measurements show some fluctuations. A sample of more than 140 water inlet temperature observations were taken. The mean of the sample inlet water temperature was 16.5℃ with a sample standard deviation of 1.93℃.

The justification of selecting the above-mentioned inputs and states is discussed in regard to field measurements. Water flow rate changes throughout plant operation from 100 to about 250 L/h. Since the heat exchanger is integrated with the process utilizing the oil, this should reflect on the return oil temperature which is the inlet to the heat exchanger. This relation is governed, however; by other than heat exchanger plant systems which may have more influencing factors. Two immanent effects are initial rate at which oil temperature decreased and the final steady temperature.

The solution methodology of the heat exchanger model considered in the present research work is based on choosing a suitable structure in the grey-box model. The structure should incorporate all inputs, outputs, and operational parameters into an appropriate mathematical formulation. In this work, the non-linear state-space structure was implemented followed by an optimization process using MATLAB identification toolbox capabilities for non-linear grey box models to estimate all unknown parameters. This process closes the model rendering it ready for performance prediction. The complete model is then coded in a MATLAB program for the calculation of the output variable against a varying range of inputs.

The suggested grey box model represents a reasonable approach in identifying a heat exchanger integrated with the oil process and could be used to implement an appropriate controller for managing process temperature. In summary, some points could be drawn from the results:

1. Increasing the cooling water flow rate can restore process temperature to its initial state but not the inlet temperature as expected.
2. The noise in inlet water temperature observations is of little significance and can be neglected.
3. Increasing oil flow rate alone to overcome elevated process temperature is not effective and should be accompanied with an adjustment to water flow rate.
4. Increasing the water flow rate alone to control the process is successful but on the expense of pumping power.
5. Thus, a control of both water and oil flow rates are mandatory for efficient temperature management.