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Chapter 6: Problem 55

Photosynthesis, as it occurs in the leaves of a green plant, involves the transport of carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) from the atmosphere to the chloroplasts of the leaves. The rate of photosynthesis may be quantified in terms of the rate of \(\mathrm{CO}_{2}\) assimilation by the chloroplasts. This assimilation is strongly influenced by \(\mathrm{CO}_{2}\) transfer through the boundary layer that develops on the leaf surface. Under conditions for which the density of \(\mathrm{CO}_{2}\) is \(6 \times 10^{-4} \mathrm{~kg} / \mathrm{m}^{3}\) in the air and \(5 \times 10^{-4} \mathrm{~kg} / \mathrm{m}^{3}\) at the leaf surface and the convection mass transfer coefficient is \(10^{-2} \mathrm{~m} / \mathrm{s}\), what is the rate of photosynthesis in terms of kilograms of \(\mathrm{CO}_{2}\) assimilated per unit time and area of leaf surface?

Short Answer

Expert verified
The rate of photosynthesis in terms of kilograms of CO2 assimilated per unit time and area of leaf surface is \(10^{-6} \mathrm{~kg} / (\mathrm{m}^{2} \cdot \mathrm{s})\).

Step by step solution

01

Write down the given values

We are given the following values: - Density of CO2 in the air: \(\rho_{\text{air}} = 6 \times 10^{-4} \mathrm{~kg} / \mathrm{m}^{3}\) - Density of CO2 at the leaf surface: \(\rho_{\text{leaf}} = 5 \times 10^{-4} \mathrm{~kg} / \mathrm{m}^{3}\) - Convection mass transfer coefficient: \(h_{m} = 10^{-2} \mathrm{~m} / \mathrm{s}\).
02

Use Fick's law of mass transfer

Fick's law of mass transfer is given by the following equation: \[ \text{Mass flux} (J) = h_{m} (\rho_{\text{air}} - \rho_{\text{leaf}}) \] Here, the mass flux (J) represents the rate of mass transfer per unit area of the leaf surface. We can plug the given values from Step 1 into this formula to find the mass flux.
03

Calculate the mass flux

Using the values of \(\rho_{\text{air}}\), \(\rho_{\text{leaf}}\), and \(h_{m}\), we can calculate the mass flux (J) as follows: \[ J = (10^{-2} \mathrm{~m} / \mathrm{s}) \times [(6 \times 10^{-4} \mathrm{~kg} / \mathrm{m}^{3}) - (5 \times 10^{-4} \mathrm{~kg} / \mathrm{m}^{3})] \] \[ J = (10^{-2} \mathrm{~m} / \mathrm{s}) \times (1 \times 10^{-4} \mathrm{~kg} / \mathrm{m}^{3}) \] \[ J = 10^{-6} \mathrm{~kg} / (\mathrm{m}^{2} \cdot \mathrm{s}) \]
04

State the rate of photosynthesis

Now that we have calculated the mass flux, we can state the rate of photosynthesis in terms of CO2 assimilated per unit time and area of leaf surface. The rate of photosynthesis is equal to the mass flux, which is: \[ \text{Rate of photosynthesis} = J = 10^{-6} \mathrm{~kg} / (\mathrm{m}^{2} \cdot \mathrm{s}) \] Therefore, the rate of photosynthesis in terms of kilograms of CO2 assimilated per unit time and area of leaf surface is \(10^{-6} \mathrm{~kg} / (\mathrm{m}^{2} \cdot \mathrm{s})\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Photosynthesis
Photosynthesis is a fundamental process that sustains plant life and, by extension, all life on Earth. It involves the conversion of light energy, usually from the sun, into chemical energy stored in glucose, a type of sugar. This process primarily occurs in the chloroplasts of plant cells, where chlorophyll, the green pigment, plays a crucial role in capturing light energy.
During photosynthesis, carbon dioxide ( CO_2 ) is absorbed from the atmosphere through small openings on leaves called stomata. Inside the chloroplasts, CO_2 combines with water ( H_2O ) absorbed by the plant's roots. This reaction, powered by sunlight, produces glucose ( C_6H_12O_6 ) and oxygen ( O_2 ) as a byproduct.
Photosynthesis can be broken down into two main stages:
  • Light-dependent reactions: These take place in the thylakoid membranes of the chloroplasts, where sunlight is absorbed and converted into chemical energy in the form of ATP and NADPH.
  • Light-independent reactions (Calvin cycle): Occurring in the stroma, these reactions use CO2 along with ATP and NADPH to synthesize glucose.
Understanding photosynthesis is crucial not just for biology students but for anyone interested in how energy flow in ecosystems sustains life.
Convection Mass Transfer Coefficient
The convection mass transfer coefficient is a key concept in understanding the rate at which a gas or liquid moves over a surface and transfers mass due to convection. It is a measure of how easily molecules transfer across the boundary layer that forms between the fluid and the surface of an object.
This coefficient, often denoted as h_m , plays a crucial role in processes such as diffusion and heat transfer. It is influenced by several factors, including:
  • Fluid properties: Viscosity and density can affect how easily molecules are moved by convection.
  • Flow conditions: Turbulence and velocity of the fluid also affect the mass transfer rate.
  • Surface type and conditions: Roughness and temperature of the surface can modify the boundary layer thickness.
The convection mass transfer coefficient is a vital parameter in fields like chemical engineering and environmental science. It helps predict and quantify the mass transfer rates necessary for processes like photosynthesis or industrial chemical reactions.
Carbon Dioxide Assimilation
Carbon dioxide assimilation refers to the process by which plants incorporate carbon dioxide into organic compounds during photosynthesis. This process is crucial because it provides the carbon necessary for the plant to synthesize glucose, which serves as energy storage and structural material.
In the context of photosynthesis, CO2 moves from the outside environment into the leaf, passing through the stomata and into the chloroplasts. This movement is influenced by the concentration gradient and the convection mass transfer coefficient, which regulate the rate of CO2 being absorbed.
Understanding carbon dioxide assimilation involves considering:
  • Diffusion: CO2 molecules move from an area of higher concentration in the air to a lower concentration inside the leaf.
  • Photosynthetic rate: The speed at which CO2 is fixed into carbohydrates directly affects plant growth and productivity.
  • Environmental factors: Light intensity, temperature, and water availability can influence how efficiently CO2 is assimilated.
Thus, understanding carbon dioxide assimilation is not only vital for appreciating the complexities of plant biology but also for addressing broader issues like agricultural productivity and carbon cycling in ecosystems.

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Most popular questions from this chapter

For laminar free convection from a heated vertical surface, the local convection coefficient may be expressed as \(h_{x}=C x^{-1 / 4}\), where \(h_{x}\) is the coefficient at a distance \(x\) from the leading edge of the surface and the quantity \(C\), which depends on the fluid properties, is independent of \(x\). Obtain an expression for the ratio \(\bar{h}_{x} / h_{x}\), where \(\bar{h}_{x}\) is the average coefficient between the leading edge \((x=0)\) and the \(x\)-location. Sketch the variation of \(h_{x}\) and \(\bar{h}_{x}\) with \(x\).

Experiments to determine the local convection heat transfer coefficient for uniform flow normal to a heated circular disk have yielded a radial Nusselt number distribution of the form $$ N u_{D}=\frac{h(r) D}{k}=N u_{o}\left[1+a\left(\frac{r}{r_{o}}\right)^{n}\right] $$ where both \(n\) and \(a\) are positive. The Nusselt number at the stagnation point is correlated in terms of the Reynolds \(\left(R e_{D}=V D / v\right)\) and Prandtl numbers $$ N u_{o}=\frac{h(r=0) D}{k}=0.814 \operatorname{Re}_{D}^{1 / 2} \mathrm{Pr}^{0.36} $$ Obtain an expression for the average Nusselt number, \(\overline{N u}_{D}=\bar{h} D / k\), corresponding to heat transfer from an isothermal disk. Typically, boundary layer development from a stagnation point yields a decaying convection coefficient with increasing distance from the stagnation point. Provide a plausible explanation for why the opposite trend is observed for the disk.

Experiments have been conducted to determine local heat transfer coefficients for flow perpendicular to a long, isothermal bar of rectangular cross section. The bar is of width \(c\) parallel to the flow, and height \(d\) normal to the flow. For Reynolds numbers in the range \(10^{4} \leq R_{d} \leq 5 \times 10^{4}\), the face-averaged Nusselt numbers are well correlated by an expression of the form The values of \(C\) and \(m\) for the front face, side faces, and back face of the rectangular rod are found to be the following: \begin{tabular}{llll} \hline Face & cld & \(\boldsymbol{C}\) & \(\boldsymbol{m}\) \\ \hline Front & \(0.33 \leq\) cld \(51.33\) & \(0.674\) & \(1 / 2\) \\ Side & \(0.33\) & \(0.153\) & \(2 / 3\) \\ Side & \(1.33\) & \(0.107\) & \(2 / 3\) \\ Back & \(0.33\) & \(0.174\) & \(2 / 3\) \\ Back & \(1.33\) & \(0.153\) & \(2 / 3\) \\ \hline \end{tabular} Determine the value of the average heat transfer coefficient for the entire exposed surface (that is, averaged over all four faces) of a \(c=40\)-mm-wide, \(d=30\)-mm-tall rectangular rod. The rod is exposed to air in cross flow at \(V=10 \mathrm{~m} / \mathrm{s}, T_{x}=300 \mathrm{~K}\). Provide a plausible explanation of the relative values of the face-averaged heat transfer coefficients on the front, side, and back faces.

On a summer day the air temperature is \(27^{\circ} \mathrm{C}\) and the relative humidity is \(30 \%\). Water evaporates from the surface of a lake at a rate of \(0.10 \mathrm{~kg} / \mathrm{h}\) per square meter of water surface area. The temperature of the water is also \(27^{\circ} \mathrm{C}\). Determine the value of the convection mass transfer coefficient. 6.53 It is observed that a 230 -mm-diameter pan of water at \(23^{\circ} \mathrm{C}\) has a mass loss rate of \(1.5 \times 10^{-5} \mathrm{~kg} / \mathrm{s}\) when the ambient air is dry and at \(23^{\circ} \mathrm{C}\). (a) Determine the convection mass transfer coefficient for this situation. (b) Estimate the evaporation mass loss rate when the ambient air has a relative humidity of \(50 \%\). (c) Estimate the evaporation mass loss rate when the water and ambient air temperatures are \(47^{\circ} \mathrm{C}\), assuming that the convection mass transfer coefficient remains unchanged and the ambient air is dry.

An industrial process involves the evaporation of water from a liquid film that forms on a contoured surface. Dry air is passed over the surface, and from laboratory measurements the convection heat transfer correlation is of the form $$ \overline{N_{L}}=0.43 \operatorname{Re}_{L}^{0.58} P r r^{\Omega .4} $$ (a) For an air temperature and velocity of \(27^{\circ} \mathrm{C}\) and \(10 \mathrm{~m} / \mathrm{s}\), respectively, what is the rate of evaporation from a surface of \(1-\mathrm{m}^{2}\) area and characteristic length \(L=1 \mathrm{~m}\) ? Approximate the density of saturated vapor as \(\rho_{A, \text { sat }}=0.0077 \mathrm{~kg} / \mathrm{m}^{3}\). (b) What is the steady-state temperature of the liquid film?
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