Question

For the case where sand is circulated between a hot utility and the pyrolysis reactor, derive an equation that gives the required sand flow (in \(\mathrm{kg}\) ) per \(\mathrm{kg}\) biomass fed as a function of the temperature differences between the pyrolysis reactor and the hot utility and the energy requirement of the reactor.

Short answer

Based on the given temperature differences between the pyrolysis reactor and the hot utility, and the energy requirement of the reactor, the required sand flow (in kg) per kg of biomass fed can be given by the following equation: \(\cfrac{m_s}{m_b} = \cfrac{Q_r}{m_b \cdot C_s(T_s - T_r)}\) Where: - \(m_s\) is the mass flow rate of sand (kg/s) - \(m_b\) is the mass flow rate of biomass (kg/s) - \(Q_r\) is the energy requirement of the reactor (J/kg) - \(C_s\) is the specific heat capacity of sand (J/kg⋅K) - \(T_s\) is the temperature of sand after gaining heat from the hot utility (K) - \(T_r\) is the temperature of the pyrolysis reactor (K)

Step-by-step solution

Identify the relevant parameters and given data

The given data in the problem consists of temperature differences between the pyrolysis reactor and the hot utility. Let's represent these as \(T_r\) (temperature of pyrolysis reactor) and \(T_h\) (temperature of hot utility). The energy requirement of the reactor is represented as \(Q_r\) (in J/kg).

Calculate the heat transfer

In order to derive an equation for the required sand flow, we need to examine the heat transfer balance between the sand, hot utility, and pyrolysis reactor. The heat transfer equation for heat exchanged between two substances can be given by: \(Q = m \cdot C \cdot \Delta T\) Where \(Q\) is the heat exchanged (J), \(m\) is the mass of the substance (kg), \(C\) is the specific heat capacity of the substance (J/kg⋅K), and \(\Delta T\) is the temperature difference (K).

Calculate heat transfer between sand and hot utility

Let's denote the mass flow rate of sand as \(m_s\) (in kg/s) and its specific heat capacity as \(C_s\) (in J/kg⋅K). The heat transfer between sand and the hot utility is given by: \(Q_s = m_s \cdot C_s \cdot (T_h - T_s)\) Where \(T_s\) is the temperature of sand (in K) after gaining heat from the hot utility.

Calculate heat transfer between sand and pyrolysis reactor

The heat transfer from the sand to the pyrolysis reactor can be determined using the same heat transfer equation: \(Q_p = m_s \cdot C_s \cdot (T_s - T_r)\) Where \(Q_p\) is the heat transfer from sand to the pyrolysis reactor (J).

Establish the relationship between heat transfers

Since the energy requirement of the reactor is given as \(Q_r\), we can establish a relationship between the heat transfers: \(Q_r = Q_p\)

Derive the equation for the required sand flow

Now, we can substitute the heat transfer equations into this relation: \(Q_r = m_s \cdot C_s \cdot (T_s - T_r)\) Divide by \(C_s\) and rearrange for \(m_s\): \(m_s = \cfrac{Q_r}{C_s(T_s - T_r)}\) Since we are asked to find the required sand flow (in kg) per kg of biomass fed, let's denote the mass flow rate of biomass as \(m_b\) (in kg/s), then the required sand flow per kg of biomass can be represented as: \(\cfrac{m_s}{m_b} = \cfrac{Q_r}{m_b \cdot C_s(T_s - T_r)}\) This equation represents the required sand flow (in kg) per kg of biomass fed as a function of the temperature differences between the pyrolysis reactor and the hot utility and the energy requirement of the reactor.

Key concepts

Pyrolysis Reactor

A pyrolysis reactor is an innovative device used to decompose organic materials at high temperatures in the absence of oxygen. It's a critical aspect of biomass energy conversion, offering a way to convert waste into biochar, bio-oil, and syngas; all valuable sources of energy. Understanding the operation of a pyrolysis reactor is fundamental when analyzing heat transfer processes.

In the context of the exercise, we're dealing with the heat transfer required to maintain the desired temperature inside the reactor. This temperature is crucial for the efficiency and effectiveness of the pyrolysis process. The sand acts as a thermal carrier, absorbing heat from a utility source and transferring it to the biomass in the reactor to trigger pyrolysis.

Biomass Energy

Biomass energy is harnessed from organic materials such as plants, agricultural waste, and wood. These resources are renewable and can be utilized to produce heat, electricity, or fuels like biodiesel and ethanol. The appeal of biomass is that it uses waste materials that are often readily available and has the potential to reduce greenhouse gas emissions compared to fossil fuels.

In the exercise, biomass is the material undergoing pyrolysis. The energy contained within the biomass is released and harnessed as a part of the biomass energy conversion process. Calculating the precise sand flow rate is essential for optimizing the energy efficiency of the pyrolysis reactor.

Specific Heat Capacity

Specific heat capacity is a property that describes how much heat energy is needed to raise the temperature of a unit mass of a substance by one degree Celsius (or one Kelvin). It's represented with the symbol 'C' and expressed in units of J/kg·K. This concept is pivotal in calculating heat transfer, as it determines how substances absorb and release heat.

The exercise specifically asks to consider the specific heat of sand because it affects the amount of energy the sand can carry to the pyrolysis reactor. Sand with a higher specific heat capacity can transfer more heat without having its own temperature rise as much, which means less sand would be required to provide the same amount of energy to the biomass.

Sand Flow Calculation

Sand flow calculation involves determining the mass flow rate of sand that's required to transfer heat at a specific rate. It's a critical aspect of processes like the operation of a pyrolysis reactor, where sand is used as a heat transfer medium. The sand must absorb heat from a utility source and then release it to the biomass, facilitating pyrolysis.

To calculate the sand flow correctly, as shown in our exercise, a thorough understanding of heat transfer principles and specific heat capacity is necessary. The derived equation takes into account the specific heat of sand, the temperature differential in the system, and the energy requirement of the pyrolysis process. This ensures that the biomass receives the right amount of energy for efficient conversion to bio-energy products.