Methane enters a 3.00-cm ID pipe at 30.0°C and 10.0 bar with an average velocity of 8.00 m/s and emerges at a point 200.0 m higher than the inlet at 30.0°C and 9.00 bar.Calculate Δ⁢E⋅k and Δ⁢E⋅p, assuming that the methane behaves as an ideal gas. **Estimate Edotk and Edotp****Note:** Check significant figures and signs throughout the calculations. Make sure that you have the correct units for R (pressures are in bar).**Calculate \( \dot{\Delta E_k} \) and \( \dot{\Delta E_p} \), assuming that the methane behaves as an ideal gas.**- \( \dot{\Delta E_k} \) : 70.36 W- \( \dot{\Delta E_p} \) : 0.2695 W**Hint:**1. Calculate the mass flow rate of methane using ideal-gas behavior.2. Calculate \( \Delta (\dot{m}gh) \).3. Calculate the exit velocity.4. Calculate \( \frac{1}{2} \dot{m} \Delta v^2 \).**Assistance Used**: Not specified.

Pipe diameter, D = 3 cm = 0.03 m Entering conditions of methane: Temperature, T1 = 30°C = 303 K Pressure, P1 = 10.0 bar The average velocity of methane in the pipe, v1 = 8.00 m/s Exit conditions of methane: Temperature, T2 = 30°C = 303 K Pressure, P2 = 9.0 bar Height of pipe at the exit, Δz = 200 mSince methane is assumed to behave as an ideal gas, the ideal gas equation is applicable to methane gas. At inlet condition, calculate the mass flow rate of methane gas as: P1V1=n˙1RT1P1v1A=m˙1MRT1 ...... (1)Here,P1=PressureV1=Volumen˙1=Molar flow rateR=Universal gas constantT1=Temperaturev1=VelocityA=area of cross-sectionm˙1=Mass flow rateM=Molecular weightSubstitute all the values as given in equation (1) and calculate the value of mass flow rate as: P1v1A=m˙1MRT110 bar8 msπ40.03 m2=m˙116 g/mol8.314×10−5 m3⋅barmol⋅K303 Km˙1=35.92 g/sm˙1=0.03592 g/s The mass flow rate of methane is the same at inlet and outlet as the flow is assumed to be at steady-state. Therefore, m˙1=m˙2=m˙=0.03592 g/s Now, the inlet and outlet temperature is the same for methane gas. Thus, T1=T2=T=303 K From the ideal gas equation applied at inlet and outlet, calculate the exit velocity of the gas as: P1V1=P2V2P1v1A=P2v2AP1v1=P2v210.0 bar8.0 ms=9.0 barv2v2=8.89 msNow, calculate the change in the kinetic energy of the methane gas as: ΔEK=m˙2v22−v12ΔEK=0.03592 kgs28.89 ms2−8.0 ms2ΔEK=0.2699 kg⋅m2s3×1 Wkg⋅m2s3ΔEK=0.2699 WThe change in the potential energy of the methane gas is calculated as: ΔEP=m˙gz2−z1ΔEP=m˙gΔzΔEP=0.03592 kgs9.81 ms2200.0 mΔEP=70.475 kg⋅m2s3×1 Wkg⋅m2s3ΔEP=70.475 W