Question

A space time of 3 hours for a flow reactor means that:

• A. The time required to process one reactor volume of feed (measured at specified conditions) is 3 hours
• B. Three reactor volumes of feed can be processed every hour
• C. After every 3 hours, the reactor needs to be cleaned
• D. The actual residence time of every fluid element is exactly 3 hours

Hint:

Definitions to remember: - Space-time (\(\tau = V / v_0\)): The time required to process one reactor volume of entering feed stream measured at baseline inlet conditions. - Space Velocity (\(s = 1 / \tau\)): The number of reactor volumes of feed that can be processed per unit time. For \(\tau = 3 \text{ hours}\), the space velocity is \(1/3\) reactor volumes per hour.

Answer 1

The Correct Option is A

Concept: Space-time (denoted by the symbol $\tau$) is a fundamental design parameter used in chemical reaction engineering to characterize the performance of continuous flow reactors, such as Continuous Stirred-Tank Reactors (CSTRs) and Plug Flow Reactors (PFRs). Space-time is defined mathematically as the ratio of the reactor volume to the volumetric flow rate of the entering feed stream: \[ \tau = \frac{V}{v_0} \] Where:
• \( V \) represents the total internal volume of the chemical reactor vessel (\(\text{m}^3\)).
• \( v_0 \) represents the volumetric flow rate of the entering feed stream (\(\text{m}^3/\text{hr}\)) evaluated at specific reference conditions (typically inlet conditions).

Step 1: Analyzing the physical definition of Space-Time.
Let us examine the physical definition of space-time by rearranging its defining equation: \[ V = \tau \cdot v_0 \] If the value of the space-time parameter ($\tau$) is given as exactly 3 hours ($\tau = 3 \text{ hours}$): \[ V = 3 \cdot v_0 \quad \Rightarrow \quad \frac{V}{v_0} = 3 \text{ hours} \] This expression states that it takes exactly 3 hours for a volume of entering feed solution equal to the total internal reactor volume ($V$) to pass completely into the system. In other words, space-time represents the time required to process one complete reactor volume of feed solution, measured at specified inlet reference conditions. This description matches option (1) perfectly.

Step 2: Differentiating Space-Time from Mean Residence Time.
It is important to distinguish space-time ($\tau$) from the actual mean residence time ($\bar{t}$) of the fluid elements inside the reactor. The mean residence time is defined as: \[ \bar{t} = \frac{V}{v_{\text{leaving}}} \] If a reaction involves a change in the total number of moles or a change in fluid density (e.g., gas-phase reactions with $\varepsilon_A \neq 0$), the volumetric flow rate will vary along the reactor length ($v \neq v_0$). In such cases, the actual residence time of a fluid element will differ from the space-time value ($\bar{t} \neq \tau$). Furthermore, in a CSTR, fluid elements exhibit a broad distribution of individual residence times, meaning that statement (4) is incorrect. Thus, statement (1) is the uniquely correct interpretation.