The output y(t) of a first-order process is governed by the following differential
equation
𝜏p
dy
dt + y = Kpf(t)
where 𝜏p is a non-zero time constant, Kp is the gain and f(t) is the input with
f(0) = 0.
Assume y(0) = 0. The transfer function for this process is (consider s as the
independent variable in the Laplace domain)
Options:
(A) Kp
𝜏ps + 1
(B) 𝜏p
Kps + 1
(C) 𝜏p
Kp(s + 1)
(D) Kp
𝜏p(s + 1)
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2025 Q.43)**
**1. Governing Concept**:
- **Topic**: Bioprocess Engineering & Continuous Thermal Sterilization
- In continuous medium sterilization (holding tube system):
- Liquid medium is rapidly heated by steam injection, passes through an insulated holding tube of length $L$ and diameter $D$, and is rapidly cooled in an expansion chamber/heat exchanger.
- Liquid velocity profile:
- Under turbulent flow ($Re > 4000$), the velocity profile is nearly flat (approximating ideal plug flow), minimizing axial dispersion.
- Under laminar flow ($Re < 2100$), the parabolic velocity profile means fluid at the centerline moves at twice the average velocity ($u_{\max} = 2 u_{\text{avg}}$), receiving only half the design residence time and leading to under-sterilization.
- Therefore, continuous holding tubes are deliberately designed to operate in the **turbulent regime** with high length-to-diameter ratio ($L/D > 100$) to minimize dispersion.
**2. Option Analysis**:
- **Option (A) is INCORRECT**: Laminar flow causes severe under-sterilization of the fastest-moving central fluid elements.
- **Option (B) is CORRECT**: Turbulent flow is essential in continuous sterilizers to ensure a narrow, uniform residence time distribution (approximating plug flow) so all fluid elements receive the required thermal exposure.
- **Option (C) is INCORRECT**: Radial temperature gradients are minimized, not maximized.
- **Option (D) is INCORRECT**: Axial dispersion must be minimized, not maximized.
**3. Final Answer**:
- **Correct Choice**: **B**
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.