An intermediate liquid product needs to be stored during 30 days at 60°C in a cylindrical tank.
A slow exothermal decomposition (400 kJ/kg) takes place with TMRad= 5days at 60°C.
Assess the thermal safety of the storage.
Data: Density = 900 kg/m3; Lambda =10 W/mk; dcrit = 2,37; Cp’ = 3,6 kJ/kgK; U = 120 W/m2K; V = 5 m3; A= 2 m2; q0 = 97 KJ/Kgs; E =10 J/mol
For the storage, the available containers are cylinders with diameters (d, cm) : 10, 15, 20, 25, 40, 50, and 100.
Which is the minimum size that would ensure safe storage.
Well, it seems like you've got quite a technical question there! Let me see if I can bring some humor into the mix while helping you out.
First, let's assess the thermal safety of the storage. To do that, we need to consider the exothermal decomposition and the temperature conditions. Think of it as a chemistry experiment gone wrong, but with a sizzling twist!
The exothermal decomposition releases 400 kJ of energy per kilogram, which means things can get pretty hot in there. Now, the TMRad of 5 days tells us that after 5 days, half of the decomposition will have occurred. That's like waiting for a pot of water to boil...except this time, it's the liquid product decomposing!
Now, let's talk about the thermal properties of the setup. The density of the liquid is 900 kg/m3, which tells me it's quite dense. Maybe it skipped leg day, but I suppose that's just its way of staying bulky!
The thermal conductivity, lambda, is 10 W/mk, which means the liquid can conduct heat pretty well. It's like the Usain Bolt of heat, always racing to spread the warmth!
Now, we need to consider the critical diameter, dcrit, which is 2.37. This is important because if the container is smaller than this critical diameter, things might go boom! It's like squeezing into your favorite pair of jeans after a month of eating nothing but ice cream. It's just not going to end well!
Moving on, we have Cp', which is the specific heat capacity, at 3.6 kJ/kgK. This tells us how much heat the liquid can absorb without getting too hot and bothered. It's like the cool friend who always knows how to handle spicy situations without breaking a sweat!
Next up, we have U, the overall heat transfer coefficient, at 120 W/m2K. This value determines how well the container can dissipate heat. Think of it as the container's ability to vent out the hot air, like a skilled ventriloquist with an endless arsenal of jokes!
We also have V, the volume of the container, at 5 m3, and A, the surface area, at 2 m2. These values tell us the size of the storage space available. It's like trying to fit a clown car full of circus performers into a tiny parking space. It's all about finding the right fit!
Lastly, we have q0, the initial heat release rate, at 97 kJ/Kgs, and E, the activation energy, at 10 J/mol. These values are like the sparks that start the fire. So, it's important to handle them with caution, just like a matchstick in a firework factory!
Now, to determine the minimum size of the container for safe storage, we need to ensure that the heat can dissipate effectively and that the temperature remains within a safe range. It's all about keeping things cool, calm, and collected, just like a laid-back surfer catching the perfect wave!
Considering all of these factors, I would recommend the 100 cm diameter container. With its larger size, it provides more surface area for heat dissipation, allowing for safer storage. It's like having a roomy mansion where everyone can cool off on a scorching summer day!
Remember, safety is key, especially when dealing with exothermal decompositions. So, always take precautions and keep things under control. Just like a clown balancing on a unicycle, it's all about finding that sweet spot of stability!
To assess the thermal safety of the storage, we need to calculate the heat generated during the 30-day storage period and compare it to the heat transfer capability of the storage container.
1. Calculate the heat generated:
The heat generated during the decomposition is given by:
q = E * (1 - exp(-t/TMRad))
Here, E is the activation energy (10 J/mol), t is the time period (30 days = 2592000 s), and TMRad is the time to maximum rate (5 days = 432000 s).
Substituting the values:
q = 10 * (1 - exp(-2592000/432000)) = 10 * (1 - exp(-6)) = 10 * (1 - 0.00248) = 9.975 J/g
Since the given data is in kJ/kg, we convert it to:
q = 9.975 * 10^-3 kJ/g * 1000 g/kg = 9.975 kJ/kg
2. Calculate the total heat generated during storage:
The total heat generated during storage is given by:
Q = q * m * V
Here, q is the heat generated per unit mass (9.975 kJ/kg), m is the density (900 kg/m^3), and V is the volume (5 m^3).
Substituting the values:
Q = 9.975 * 900 * 5 = 44887.5 kJ
3. Calculate the heat transfer capability of the storage container:
The heat transfer capability is given by:
Q' = U * A * ΔT
Here, U is the overall heat transfer coefficient (120 W/m^2K), A is the heat transfer area (2 m^2), and ΔT is the temperature difference (60°C).
Converting the temperature to Kelvin:
ΔT = 60 + 273.15 = 333.15 K
Substituting the values:
Q' = 120 * 2 * 333.15 = 79836 W = 79.836 kJ
4. Compare the heat generated and the heat transfer capability:
If the heat generated (Q) is greater than the heat transfer capability (Q'), it indicates that the storage is not thermally safe.
Comparing the values:
Q = 44887.5 kJ
Q' = 79.836 kJ
Since Q > Q', the storage is not thermally safe.
To determine the minimum size of the container that would ensure safe storage, we need to find the maximum allowable heat generation per unit volume.
5. Calculate the maximum allowable heat generation per unit volume:
To ensure thermal safety, the heat generation per unit volume should be less than the critical heat generation per unit volume (dcrit).
The maximum allowable heat generation per unit volume is given by:
q_max = dcrit * E
Here, dcrit is the critical heat generation per unit volume (2.37) and E is the activation energy (10 J/mol).
Substituting the values:
q_max = 2.37 * 10 = 23.7 J/g
Since the given data is in kJ/kg, we convert it to:
q_max = 23.7 * 10^-3 kJ/g * 1000 g/kg = 23.7 kJ/kg
6. Calculate the minimum container size:
The minimum container size can be determined by finding the maximum allowable heat generation per unit volume.
The volume of the container is given by:
V_min = Q/q_max
Here, Q is the total heat generated during storage (44887.5 kJ) and q_max is the maximum allowable heat generation per unit volume (23.7 kJ/kg).
Substituting the values:
V_min = 44887.5 / 23.7 = 1892.52 kg
Since the given data is in m^3, we convert it to:
V_min = 1892.52 * 10^-3 m^3 = 1.89252 m^3
Therefore, the minimum container size that would ensure safe storage is approximately 1.89252 m^3.
To assess the thermal safety of the storage, we need to calculate the heat generation rate due to the exothermal decomposition and compare it with the heat dissipation rate given the conditions and parameters provided.
To calculate the heat generation rate due to exothermal decomposition, we can use the formula:
Qgen = (q0 * m * f(T) * t) / M
Where:
- Qgen is the heat generation rate (in Watts)
- q0 is the heat release per unit mass (in kJ/kg)
- m is the mass of the substance (in kg)
- f(T) is the temperature correction factor
- t is the time (in seconds)
- M is the molar mass of the substance (in kg/mol)
The temperature correction factor, f(T), can be calculated using the equation:
f(T) = exp(-E/RT)
Where:
- E is the activation energy (in J/mol)
- R is the ideal gas constant (8.314 J/mol·K)
- T is the temperature (in Kelvin)
Given:
- q0 = 97 kJ/kg
- E = 10 J/mol
- T = 60°C = 333 K
First, let's calculate the temperature correction factor, f(T):
R = 8.314 J/mol·K
E = 10 J/mol
T = 333 K
f(T) = exp(-10 / (8.314 * 333))
f(T) ≈ 0.99962
Now, we can calculate the heat generation rate, Qgen. But first, we need to calculate the mass of the substance, m, using the density and volume:
Density = 900 kg/m3
Volume = 5 m3
m = Density * Volume
m = 900 * 5
m = 4500 kg
Also, we need to convert the time from days to seconds:
t = 30 days * 24 hours/day * 60 minutes/hour * 60 seconds/minute
t ≈ 2,592,000 seconds
Now, we can calculate Qgen:
Qgen = (q0 * m * f(T) * t) / M
Given:
- q0 = 97 kJ/kg
- m = 4500 kg
- f(T) ≈ 0.99962
- t ≈ 2,592,000 seconds
- M (molar mass) is not given
Unfortunately, the molar mass (M) of the substance is not provided in the given data. Without the molar mass, we are unable to calculate the heat generation rate due to exothermal decomposition (Qgen) accurately.
However, if we assume a typical molar mass for an organic compound, such as 100 g/mol, we can continue the calculation for the heat dissipation rate to determine the minimum size of the cylinder for safe storage.
To calculate the heat dissipation rate, we can use the formula:
Qdiss = U * A * (T - Tref)
Where:
- Qdiss is the heat dissipation rate (in Watts)
- U is the heat transfer coefficient (in W/m2K)
- A is the surface area of the cylinder (in m2)
- T is the temperature of the stored liquid (in °C)
- Tref is the reference temperature (in °C), typically the ambient temperature
Given:
- U = 120 W/m2K
- A = 2 m2
- T = 60°C
- Tref (ambient temperature) is not provided
Unfortunately, the reference temperature (Tref) is not provided, and it is essential to accurately calculate the heat dissipation rate (Qdiss) for determining the minimum safe size of the cylinder.
Without additional information, it is not possible to complete the assessment of the thermal safety of the storage or determine the minimum size of the cylinder for safe storage.