## October 2019

## Process Optimization

# A dynamic 1D model for accelerated reactor cooling

Refinery hydrofining units must undergo planned and emergency shutdowns for catalyst changeover.

Refinery hydrofining units must undergo planned and emergency shutdowns for catalyst changeover. Among the many pre-shutdown operations—including catalyst washing, inertization of pyrophoric chemicals, hydrocarbon stripping, reactor cooling and atmospheric inertization—the reactor cooling process is particularly lengthy.

Reactors are cooled from their working temperature (400°C–450°C) to approximately 130°C–150°C by utilizing the high-pressure, hydrogen (H_{2})-rich gas recycle loop as the air-cooled exchanger—usually present in the loop. This is sufficiently effective in the cooling gas leaving the reactor bottoms. This phase is also utilized by operators to comply with the minimum pressurization temperature (MPT) cooling curve of the specific reactor. Cooling reactors from 130°C to 30°C–35°C is particularly challenging due to the reduced *ΔT* approach at the air cooler and because of the thermal power introduced by the recycling compressor into the cooling gas.

Depending on the reactor size, this cooling phase may last a couple of weeks—if counting on natural cooling only—or 4 d–4.5 d when nitrogen injection is provided at the bottom of the reactor in a once-through configuration.

Reactor cooling time can be reduced by providing external refrigeration from the circulated H_{2}-rich gas. The recirculation is provided by the recycle compressor, while temporary chilling units and exchangers ensure refrigeration of the gas that is recycled to the reactor, thereby shortening the cooling time to a few hours.

The aim of this article is to present a mathematical model (**Fig. 1**) to predict the reactor cooling time as a function of the recycled gas pressure, temperature and flowrate. A one-dimensional (1D) dynamic cooling model applied to refinery reactors allows for easy comparison of cooling times. Its simplicity and flexibility are utilized for different refrigeration techniques and fluids.

## Thermodynamic model

The thermodynamic model is built under a few simplifying assumptions:

- All reactor weight is distributed on the cylindrical part; this includes internal, top and bottom heads
- All catalyst weight is uniformly distributed along the reactor height
- The gas inside the reactor is assumed in plug flow
- The reactor does not release any energy to the ambient atmosphere
- Calculations are performed considering average properties of the gas, catalyst and metal
- The heat transfer coefficient of the gas phase is assumed to be constant along the reactor height
- Heat transfer between the catalyst and reactor vessel is neglected
- No axial or radial heat transport is assumed.

## Material and energy balance

The thermodynamic model under the outlined assumptions is represented by Eqs. 1–10, specifically Eqs. 7–10:

Heat released by metal phase:

(1)

Heat exchanged on metal-gas interface:

(2)

Heat released by catalyst phase:

(3)

Heat exchanged on catalyst-gas interface:

(4)

Heat received by gas phase:

(5)

Energy conservation:

*d**Q _{G }*

*= d*

*Q*

_{m }*+ d*

*Q*

_{c}*[J] (6)*

Eqs. 1–6 can be presented in a more convenient form, as shown in Eqs. 7–10:

(7)

(8)

(9)

(10)

By combining and deriving Eqs. 7–10, we can reduce the system to Eq. 11:

(11)

The coefficients are defined in the following:

Applying the separation of variables in Eq. 11, *T _{G}(z,t) = T(t) × T(z),* as shown in Eq. 12:

(12)

With *T(t)'*, *T(t)''*, *T(t)'''* first-, second- and third-time derivatives, we can calculate Eqs. 13 and 14:

*T(t)''' + a'T(t)'' + b' T(t)' + c' T(t)* = 0 (13)

*T(z)' – λT(z) = 0 *(14)

The constants are defined in the following:

The solutions of Eqs. 13 and 14 are shown in Eq. 15:

(15)

To obtain acceptable solutions,* k _{1}* and

*k*must be real and negative for

_{2}*λ*to be real and positive.

When Eq. 15 is plugged into Eqs. 7–10, equivalent expressions for *T _{m}* and

*T*can be calculated as shown in Eqs. 16 and 17:

_{c}(16)

(17)

With the constants defined as:

Finally, Eq. 18 can be calculated:

(18)

Eqs. 15–17 are used to calculate the temperature profile along the reactor axis, starting with shell and catalyst initial temperatures *T _{mi}* at any instant between

*t*= 0 and

*t,*with

*t*being the time necessary to cool the reactor bottom shell below the target temperature (in the following examples, this is assumed to be 40°C). Eq. 18 is used to close the thermal balance along the reactor height.

## Case study

A case study describing the cooling of a reactor is presented here. **Fig. 2** represents one of the many possible configurations to perform accelerated reactor cooling with external refrigeration. The reactor presents the following characteristics:

- Diameter: 3,500 mm
- Height: 30,000 mm
- Shell thickness: 147 mm
- Weight of shell and internals: 407 t
- Catalyst weight: 150 t
- Initial temperature: 120°C
- Final desired temperature: 40°C
- The reactor had been previously cooled utilizing 9 tph of once-through nitrogen gas (N
_{2}) at a low pressure of 2.2 barg.

Due to the large available external surface area per kg of catalyst, the heat transfer rate from catalyst to the gas phase is significant, so parameters like bed porosity and external surface per m^{3} have been roughly estimated in *ε* = 20% and *a _{v}* = 10m

^{2}/m

^{3}, as the total cooling time of the reactor has a low sensitivity to changes in catalyst parameters.

Furthermore, the catalyst bed at its end of life can be lumped in blocks not permeable to the gas; preferential paths and bypass zones can also exist. No further accuracy is needed to define the catalyst bed in the limits of the initial assumption of the cooling model. Thermal conductivity and bed density have been estimated at *k* = 10 W/mK and *ρ* = 900 kg/m^{3}.

The model has been calibrated to the specific reactor system, under the knowledge that the desired cooling was achieved in 5 d (specifically, 113 hr) using N_{2} (see **Fig. 3 **for calibrating simulations). The average gas phase heat transfer coefficient (assumed to be *U* = 50 W/m^{2}K) has been used as a calibration parameter for the model. The reactor shell was considered to be thermally equivalent to stainless steel AISI 316.

Accelerated cooling with H_{2}-rich gas. The high-pressure recycle loop was operated with H_{2}-rich gas as cooling medium, injected at the top end at 15°C and 35 barg. As the compressor was operated at 2,340 m^{3}/hr and a power of 222 kW, 7 tph was the calculated H_{2} mass flowrate.

As the theoretical gas heat transfer coefficient of the H_{2} at 35 barg is about 6.8 times higher than the theoretical heat transfer coefficient in the N_{2} at 2.2 barg, the cooling time simulation in H_{2} at 35 barg was performed assuming *U* = 339 W/m^{2}K. This resulted in a total cooling time of 14 hr (**Fig. 4**).

Cooling operations were not conducted in a continuous manner, and the H_{2} gas had variable characteristics during the cooling phase. The total cooling time was within 15% of the simulated time frame.

Accelerated cooling with N_{2} circulation. The same model was applied to the high-pressure recycle loop operated with N_{2} as cooling medium, injected at the top end at 15°C and 35 barg. The volumetric gas flowrate at the compressor was the same used in the H_{2} cooling case, with a mass flowrate of 98 tph.

Utilizing *U* = 50 W/m^{2}K as the calibration parameter, the simulated cooling time was found to be 18 hr (**Fig. 5**).

Accelerated cooling with bottom-up water circulation. Cooling of the reactor can also be performed with water having an initial temperature of 90°C and a final desired temperature of 30°C. The water flowrate utilized was 200 m^{3}/hr injected from the bottom in a single point and circulated from the reactor top through a water chiller, thereby refrigerating the water at 5°C.

The total simulated cooling time of 7 hr and 45 min. is in good agreement with recorded field data (**Fig. 6**).

## Takeaway

Comparing results—14 hr in an H_{2} atmosphere and 18 hr in an N_{2} atmosphere (only theoretical)—with nearly atmospheric N_{2} once-through cooling, the difference in cooling time shows a gain of approximately 95 hr of reactor production (approximately 4 d), while avoiding consumption of nearly 9 tph of liquid nitrogen.

Such a high volumetric nitrogen flowrate creates pressure drops in the flare header, with consequent increased backpressure and built-up backpressures on the safety valves connected. This may limit relieving capacity of these safety devices. Furthermore, to sustain flammability of the gases and vapors discharged into the flare system, some fuel gas must be injected together with the discharged nitrogen during cooling operations.

The 1D dynamic cooling model was calibrated, using the available data previously recorded during low-pressure cooling in an N_{2} atmosphere. After pro-rating the average heat transfer coefficients for H_{2} atmosphere at 35 barg, the simulation of the total cooling time (14 hr) was then performed. The results were within 15% of the calculated time. The same simulation was then conducted, using high-pressure circulation of N_{2}. This resulted in a cooling time of 18 hr.

The dynamic 1D cooling model allows realistic predictions of the reactor cooling time with different gases, as well as with water flood techniques, by adjusting the gas phase heat transfer coefficient, bed porosity and catalyst physical properties. **HP**

**Nomenclature**

For shell:

*D _{i}* Reactor internal diameter, m

*H _{r}* Reactor height total length to total length, m

*W _{m}* Shell total weight distributed on the cylindrical shape, kg

*ρ _{m}* Metal density, kg/m

^{3}

*k _{m}* Metal thermal conductivity, W/mK

*cp _{m}* Heat capacity metal, J/kgK

*(T _{m})_{z}* Metal temperature at

*z*coordinate, °C

For catalyst:

*W _{c}* Catalyst total weight uniformly distributed on catalyst bed height, kg

*ρ _{c}* Catalyst bulk density, kg/m

^{3}

*k _{c}* Catalyst thermal conductivity, W/mK

*cp _{c}* Catalyst heat capacity, J/kgK

*ε* Catalyst bed porosity, %

*a _{V}* Volume surface area, m

^{2}/m

^{3}

*(T _{c})_{z}* Catalyst temperature at z coordinate, °C

For gas:

*W _{G}* Gas mass flowrate, kg/sec

*ρ _{G}* Gas density, kg/m

^{3}

*k _{G}* Gas thermal conductivity, W/mK

*cp _{G}* Gas heat capacity, J/kgK

*(T _{G})_{z}* Gas temperature at

*z*coordinate, °C

*(T _{Gi})_{z=0}* Gas temperature at reactor top, °C

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