Let's solve the problem step by step, with explanations at each stage. --- ### **Step 1: Draw and Label the Diagram** #### **Description of the Setup** - A rectangular tank (length \( a \), breadth \( b \)) is filled with fluid to depth \( d \), fluid density \( R \). - Above the fluid is inert gas at absolute pressure \( p \) (in kPa). - A 50 mm diameter pipe is attached to the tank bottom. - A 25 mm diameter branch pipe connects to the offtake pipe \( h \) meters below the tank bottom. - The 25 mm pipe runs horizontally, then rises at angle \( f \) (in radians) for hypotenuse \( k \) (in cm), then rises vertically for \( v \) (in mm). - Define \( z_1 \) as the elevation at the fluid/gas interface (top surface of the liquid in the tank). - Define \( z_2 \) as the elevation at the end of the 25 mm branch pipe. - **Ignore pipe diameters for elevations.** #### **How to Draw** 1. Draw the tank, with top surface labeled \( z_1 \). 2. Show the bottom of the tank. 3. Draw a pipe descending \( h \) below the tank bottom. 4. Draw the branch pipe: - Horizontal section, - Slanted up section at angle \( f \) (length \( k \) cm), - Vertical up section (height \( v \) mm). 5. The end of the branch is at \( z_2 \). **(You are instructed to draw this yourself and upload a photo. The diagram cannot be created in text here, but the above description allows you to sketch it accurately.)** --- ### **Step 2: Find the Total Vertical Height Difference between \( z_1 \) and \( z_2 \)** #### **Explanation** To find the static pressure at the branch pipe's end, we need the vertical distance between \( z_1 \) (top of liquid) and \( z_2 \) (end of pipe). Starting from \( z_1 \): - To tank bottom: **down \( d \) m** - To branch tapping: **down \( h \) m** - 25 mm pipe: - The horizontal section does **not** change elevation. - The slanted section rises: vertical rise = \( k \sin(f) \) (since \( k \) is hypotenuse, \( f \) is angle above horizontal). - Remember to convert \( k \) from cm to m: \( k/100 \) m. - The vertical section rises: \( v \) mm \( = v/1000 \) m. So, **total vertical drop from \( z_1 \) to \( z_2 \):** \[ z_1 - z_2 = d + h - (k/100) \sin(f) - (v/1000) \] --- ### **Step 3: Write the Static Pressure at \( z_2 \) using Hydrostatics** #### **Explanation** The absolute pressure at \( z_2 \) is: - The gas pressure at the top \( (p) \) (convert from kPa to Pa: \( p \times 1000 \)). - Plus the hydrostatic pressure due to the column of fluid of height \( z_1 - z_2 \) and density \( R \), i.e., \( R g (z_1 - z_2) \). So, \[ P_{z_2} = (p \times 1000) + R g (z_1 - z_2) \] Substitute from Step 2: \[ P_{z_2} = (p \times 1000) + R g \left[d + h - \frac{k}{100} \sin(f) - \frac{v}{1000}\right] \] --- ### **Step 4: Final Formula and Explanation** #### **Explanation** This formula gives the absolute static pressure at the end of the 25 mm branch pipe, in Pascals, using all the given variables and proper unit conversions. --- ### **Final Answer** \[ \boxed{ P_{z_2} = (p \times 1000) + R\,g\,\left[d + h - \frac{k}{100} \sin(f) - \frac{v}{1000}\right] } \] - \( p \) = pressure above fluid (in kPa) - \( R \) = fluid density (kg/m³) - \( g \) = gravitational acceleration (m/s²) - \( d \) = fluid depth in tank (m) - \( h \) = vertical distance below tank bottom to branch (m) - \( k \) = length of slanted branch (cm) - \( f \) = angle of slant (radians) - \( v \) = vertical branch rise (mm) **This formula uses only the symbols as defined, with correct unit conversions.**