Stuck on this homework problem, any help would be appreciated. Thank you!!
A truck has 5.9L diesel engine in it. 5.9L of the overall displacement in 6 cylinders each 102mm bore (diameter) and 120mm stroke (length). Engine operates at 16.3:1 compression ratio and with 6psi (0.41atm) of boost cruising at 1600rpm.
1.) The intake valve closes and the piston compresses the air/fuel mixture to 1/16.3=6% of the initial volume. Calculate the new volume and pressure.
(for the initial volume I got 1,039,387.133mm^3)
2. Diesel engine have high thermal efficiency than gasoline engines, 45% efficient. Thus 45% of the energy in part G is work W and 55% of the energy is heart Q. Calculate W and Q.
3. Use W and the cylinder volume to calculate an average pressure in the cylinder during the combustion stroke (101.3 J= 1L atm)
1. To calculate the new volume and pressure after the piston compresses the air/fuel mixture, follow these steps:
Step 1: Calculate the initial volume:
Given the initial bore diameter (d1) of 102mm and stroke length (l) of 120mm, we can find the initial volume (V1) using the formula for the volume of a cylinder:
V1 = π/4 * d1^2 * l
Substituting the values, we get:
V1 = π/4 * (102mm)^2 * 120mm
Step 2: Calculate the new volume:
The air/fuel mixture is compressed by a ratio of 1/16.3. Therefore, the new volume (V2) can be calculated by multiplying the initial volume (V1) by 1/16.3:
V2 = V1 * (1/16.3)
Step 3: Calculate the new pressure:
Given that the engine operates at 6 psi of boost, we need to convert it to atmospheres (atm):
1 atm = 14.7 psi (approximately)
So, the boost pressure in atmospheres is:
P_boost = 6 psi / 14.7 atm/psi
To find the new pressure (P2), we can use Boyle's Law, which states that the product of pressure and volume is constant:
P1 * V1 = P2 * V2
Rearranging the equation to solve for P2:
P2 = (P1 * V1) / V2
Substituting the values, we get:
P2 = (P_boost * V1) / V2
Calculate the values of V1 and V2, then substitute them into the equation to find the new pressure P2.
2. To calculate the work (W) and heat (Q) in part G, follow these steps:
Given that the diesel engine has a thermal efficiency of 45%, we know that 45% of the energy is work, and 55% is heat.
Step 1: Calculate the total energy in part G (E_total):
E_total = W + Q
Step 2: Convert the efficiency to a decimal:
Thermal efficiency = 45% = 0.45
Step 3: Set up an equation using the efficiency equation:
E_total = E_input * Efficiency
Step 4: Rearrange the equation to solve for W:
W = E_total * Efficiency
Step 5: Substitute the values into the equation to calculate W.
Once you have calculated W, you can calculate Q by subtracting W from E_total.
3. To calculate the average pressure in the cylinder during the combustion stroke, follow these steps:
Step 1: Convert the work (W) from the previous question to joules (J) if it is not already in joules.
Step 2: Calculate the change in volume during combustion using the formula:
ΔV = V1 - V2
Step 3: Convert the cylinder volume from mm^3 to L:
Cylinder volume = V1 (from question 1) / 1000
Step 4: Calculate the average pressure (P_avg) using the formula:
P_avg = W / ΔV
Step 5: Convert the pressure to atm if needed.
Substitute the values into the equation to calculate the average pressure in the cylinder during the combustion stroke.
1. To calculate the new volume and pressure after compressing the air/fuel mixture to 6% of the initial volume, you can use the formula:
V2 = V1 * (1 / compression ratio)
Where:
V2 is the new volume after compression
V1 is the initial volume before compression
Compression ratio is given as 16.3:1 (which can also be written as 16.3)
Let's calculate the new volume (V2):
V2 = 1,039,387.133 mm^3 * (1 / 16.3)
V2 ≈ 63,723.158 mm^3
Now, to calculate the new pressure, you can use Boyle's Law, which states that the pressure and volume of a gas are inversely proportional, given a constant temperature.
P1 * V1 = P2 * V2
Where:
P1 is the initial pressure (which we don't know yet)
P2 is the new pressure (what we want to calculate)
V1 is the initial volume (1,039,387.133 mm^3)
V2 is the new volume (63,723.158 mm^3)
Since we don't know P1, we'll rearrange the formula to solve for P2:
P2 = (P1 * V1) / V2
Now, we need to convert the volume from mm^3 to liters (L), and also convert the given boost pressure from psi to atm.
V1 = 1,039,387.133 mm^3 / 1,000,000 = 1.039387133 L
boost pressure = 6 psi * 0.068046 = 0.408276 atm
Substituting the values into the formula:
P2 = (P1 * 1.039387133 L) / 0.063723158 L
Simplifying the equation, we get:
P2 = P1 * 16.285633231
This equation tells us that the new pressure (P2) is 16.285633231 times greater than the initial pressure (P1). However, we don't have enough information to calculate the actual pressure. If you have the initial pressure (P1), you can substitute it into the equation to calculate the new pressure (P2).
2. To calculate the work (W) and heat (Q) energy in the engine, you can use the efficiency of the diesel engine. Given that the engine is 45% efficient:
Efficiency = W / (W + Q)
Rearranging the equation, we can solve for W:
W = Efficiency * (W + Q)
Substituting the given efficiency of 45% (0.45):
W = 0.45 * (W + Q)
Expanding the equation gives:
W = 0.45W + 0.45Q
Now, if we rearrange it to isolate W:
W - 0.45W = 0.45Q
0.55W = 0.45Q
Dividing both sides by 0.55:
W = (0.45Q) / 0.55
This equation gives the value of the work (W) as a fraction of the heat (Q). However, we still need the value of Q to calculate W. Without additional information regarding Q, it is not possible to calculate the specific values for W and Q.
3. To calculate the average pressure in the cylinder during the combustion stroke using work (W) and cylinder volume, you can use the formula:
Pressure = (W / Cylinder Volume) * 101.3 J / 1L atm
Substitute the given values for W and the cylinder volume to calculate the average pressure.