Transient Heat Transfer Experiment Report

Introduction

Learning Outcomes:

The primary objective of this experiment is to gain an understanding of transient heating and cooling in a tank containing an aqueous solution. Additionally, we aim to familiarize ourselves with the WL110 Software, which is used to log data and analyze it. The experiment involves calculating the rate of heat loss from the hot fluid in the heat jacket to the cooling water inside the tank. Lastly, we will identify and address any potential hazards associated with the experiment.

Theory

Heating a well-mixed batch of liquid in a tank is accomplished by passing hot water around the fluid. This process facilitates heat transfer from high-energy particles to lower-energy particles. Consider a liquid with a mass (M) in kilograms, initially at a temperature (T0) in Kelvin, and a constant flow rate (T1) in liters per minute into the jacket. The equation for the heat energy (Q) required for heat transfer is given by:

Q = M * cp * (T - T0)       (1)

Where:

• cp (J kg^-1 K^-1) is the specific heat capacity of the fluid in the tank.
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• T is the final temperature of the fluid in Kelvin.

The rate of change in energy can be calculated by differentiating Equation (1) with respect to time (t), resulting in:

dQ/dt = M * cp * dT/dt       (2)

The rate of energy loss of the hot fluid in the fluid jacket can be expressed as:

dQ/dt = UA * (Th - T)       (3)

Where:

• U (W m^-2 K^-1) is the overall heat transfer coefficient.
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• A (m²) is the heat transfer area.
• Th is the initial temperature of the hot fluid.

Using Equations (2) and (3), the rate of energy gained by the fluid in the tank is equal to the rate of energy lost by the hot fluid in the jacket, resulting in the equation:

UA * (Th - T) = M * cp * dT/dt       (4)

For convenience, we replace A with Am, the logarithmic mean heat transfer area, defined as:

Am = (A_out - A_in) / ln(A_out / A_in)       (5)

We also introduce ∆Th,m to account for fluctuations, where ∆Th,m is the logarithmic mean temperature difference between the inlet and outlet temperatures:

∆Th,m = (∆Th,max - ∆Th,min) / ln(∆Th,max / ∆Th,min)       (6)

Here, ∆Th,max and ∆Th,min represent the largest and smallest differences between the inlet and outlet flow temperatures, respectively.

If Th,in is not constant, an average must be taken between all the inlet flow temperatures to obtain a mean value. The mean outlet temperature of the fluid in the jacket, denoted as ¯T, is calculated as:

¯T_(h,out) = Th,in - ∆Th,m       (7)

The logarithmic mean temperature of the fluid is then computed using:

T_(h,m) = (Th,in - ¯T_(h,out)) / ln(Th,in / ¯T_(h,out))       (8)

Substituting Am and Th,m into Equation (4), we arrive at the following expression:

(T_(h,m) - T(t)) / (T_(h,m) - T0) = exp⁡[-(UA_m) / (M * cp) * t] = e^(-t / τ)       (9)

Where:

• τ = (M * cp) / (UA_m)       (10)
• T(t) indicates that temperature T is a function of time.

Relevance

Heat transfer plays a crucial role in various industries, including the food and pharmaceutical sectors, which involve processes containing enzymes. Maintaining constant temperatures is essential for enzymes to operate efficiently, as extreme temperatures can either slow down enzyme reactions or denature the enzymes themselves. Transient heat transfer is a valuable technique for ensuring a consistent temperature, thereby promoting a high rate of enzyme reaction by cooling or heating a fluid as needed.

Experimental Work

System Used

In this experiment, water was used as both the fluid inside the tank and in the fluid jacket. The heat exchanger was constructed from stainless steel.

Equipment and Procedures

The following equipment and procedures were employed:

1. Utilized the GUNT Heat Exchanger Service Unit WL 110, which included a tank for supplying hot water via a boiler.
2. The unit was equipped with thermocouples, T1 (Hot Inlet), T2 (Heat Exchanger Vessel), and T3 (Hot Outlet), which measured temperatures and logged data using the WL110 Software on a computer.
3. The Heat exchanger vessel, WL 110.04, featured an agitator, which operated at maximum speed to ensure uniform heat distribution and facilitate one-dimensional heat transfer.
4. Set the set point on C7 Controller to 70˚C and turned on the heater.
5. Adjusted the inlet flow to 2.0 liters per minute using valve Vh.
6. Weighed out 1.2 kg of cold water in a beaker and recorded the mass on a balance.
7. Waited for the inlet temperature (T1) to reach a constant 70˚C and initiated data recording on the WL110 software while pouring the measured water into the tank and turning the agitator to its maximum speed.
8. Stopped recording when the temperature in the heat exchanger vessel (T2) reached a steady state.
9. Repeated the above steps with an inlet flow rate of 1.0 liter per minute.

Hazards

One of the hazards in this experiment was the use of hot water in the pipes, which could cause harm on contact. Proper precautions, such as applying cold water to burns to prevent scalds and blisters, were taken to ensure safety during the experiment.

Results

To calculate the height (h) of the cylinder, we assumed that the volume of water was equal to the volume of the tank. The volume equation for a cylinder is given by:

V"tank" = π * r^2 * h       (11)

We also calculated the volume of water using mass (M) and density (ρ):

V"water" = M / ρ       (12)

With the mass of 1.1985 kg of water and a known density of 997 kg m^-3, we calculated hin and hout for different tank diameters using Equation (11).

The surface area of a cylinder can be determined using the formula:

A = 2 * π * r * h + 2 * π * r^2       (13)

Utilizing Equation (5), we calculated Am using Ain and Aout from Equation (14):

Am = (0.163326 - 0.144281) / ln(0.144281 / 0.163326)       (14)

We also calculated ∆Th,m using Equation (6) for the flow rate of 2.0 liters per minute:

∆Th,m = (∆Th,max - ∆Th,min) / ln(∆Th,max / ∆Th,min)       (15)

Experimental Data

Flow rate (l/min)	Mass of Water used (kg)	Am (m2)	T0 (K)	Th, m (K)	Th, in (K)	¯Th, out (K)	∆Th, m (K)	τ (s)	U (W m-2 K-1)	R (K W-1)	R (%)
1.0	1.1985	0.153607	308.7168	342.5102	342.8952	340.8237	2.3494	1111.1111	75.421	0.001525	1.28
2.0	1.2004	0.155203	304.5693	343.5710	343.9762	342.2535	1.0755	344.8276	233.326	0.000857	4.02

Discussion

Analysis of the results reveals that a higher flow rate of 2.0 liters per minute leads to the system reaching a steady state in less time, resulting in larger increases in temperature over time. This observation aligns with Equation (3), where the rate of heat transfer is directly proportional to the difference in temperature. Additionally, the first law of thermodynamics dictates that T2 should never exceed T1 in normal circumstances, which is evident in the data. The trend in the graphs corresponds to the theoretical expectations.

The agitator's role in the experiment is significant as it ensures the uniform distribution of energy. The graphs do not start from the point (0,0) because the heat exchanger was not initially in a stable state until the agitator achieved uniform heat distribution. The use of a data logger recording temperatures at one-second intervals was crucial, as manual recording could introduce human errors when reading thermocouples. It's important to note that our calculations may not be precise due to the standard error of the balance used to weigh the water, with an uncertainty of ±0.05g, impacting the measurements.

Conclusion

In conclusion, this experiment demonstrates that increasing the flow rate of the hot fluid results in a shorter time required to heat the tank to a specific temperature. As evidenced by Equation (3), the rate of heat transfer is directly influenced by the temperature difference. This finding is consistent with theoretical expectations and supports the hypothesis that a higher flow rate accelerates the heating process.

Nomenclature

The following nomenclature was used throughout the experiment:

• Q: Heat required (J)
• cp: Specific Heat Capacity (kJ kg^-1 K^-1)
Din: Inner diameter of the tank (m)
• Dout: Outer diameter of the tank (m)
• U: Overall Heat Transfer Coefficient (W m^-2 K^-1)
• Ain: Inner Surface Area of the Tank (m²)
• Aout: Outer Surface Area of the Tank (m²)
• Am: Surface Area of the Tank (m²) - Logarithmic mean transfer area
• M: Mass (kg)
• ρ: Density of fluid (kg m^-3)
• T: Final Temperature (K)
• T0: Initial Temperature (K)
• dQ/dt: Rate of change of heat energy gained/lost with respect to time (J per unit time)
• dT/dt: Rate of change of temperature with respect to time (K per unit time)
• Th: Initial Temperature of hot fluid (K)
• ∆Th,max: Max difference between inlet and outlet temperature (K)
• ∆Th,min: Minimum difference between inlet and outlet temperature (K)
• ∆Th,m: Logarithmic mean temperature difference between inlet and outlet temperature (K)
• T_(h,out): Mean outlet temperature of the outlet (K)
• T_(h,m): Logarithmic mean temperature of the fluid in the jacket (K)
• T(t): Temperature as a function of time (K)
• τ: Time Constant (s)
• Vin: Volume of inner cylindrical tank (m³)
• Vout: Volume of outer cylindrical tank (m³)
• T2: Temperature of fluid inside the tank (K)
• T3: Temperature of the outlet flow of the jacket (K)
• R: Overall resistance to heat transfer (K W^-1)
• R_tank: Heat resistance from tank wall (K W^-1)
• x: Thickness of the tank wall (m)
• k: Thermal Conductivity of the Tank wall (W m^-1 K^-1)
• R%: Contribution of heat resistance from the tank wall to the overall heat resistance as a percentage (%)

This comprehensive experiment report provides valuable insights into transient heat transfer and its practical applications. Understanding heat transfer processes is essential in various industries, and the results obtained in this experiment contribute to this knowledge.