Heating, Ventilation, and Air Conditioning

Abstract

The heating, ventilation and air conditioning (HVAC) challenges faced in the food-manufacturing industry are: to ensure that the employees benefit from a safe, comfortable, and productive working environment; and to facilitate the manufacture of consistently safe and high-quality products by ensuring that the activities are undertaken in hygienic and temperature/humidity controlled conditions.

Appendices

Appendix 1: Derivation of Transient Temperature, Humidity, and Pollutant Concentration Equations

Transient Temperature

Consider a room of volume V, with an air exchange rate of Q m³/s. An enthalpy balance on the room yields:

$$ \mathrm{ Rate}\ \mathrm{ of}\ \mathrm{ change}\ \mathrm{ of}\ \mathrm{ enthalpy}=\mathrm{ Heat}\ \mathrm{ in}/\mathrm{ out}+\mathrm{ Energy}\ \mathrm{ generation}+\mathrm{ Heat}\ \mathrm{ transfer}\ \mathrm{ to}\ \mathrm{ room}. $$

$$ \rho V\mathrm{ Cp}\frac{{\mathrm{ d}T}}{{\mathrm{ d}t}}=\rho Q\mathrm{ Cp}\left( {{T_{\mathrm{ a}}}-T} \right)+W+UA\left( {{T_{\mathrm{ a}}}-T} \right), $$

(16.26)

where T and T _a are the temperatures (K) of the room and outside air respectively, t is time (s), Cp and ρ are the specific heat (J/kgK) and density (kg/m³) of the air, W is the rate of heat generation inside the room (W), U is the effective heat transfer coefficient (W/m²K) between the air in the room and that outside, and A is the effective coupling area (m²) between the room and outside. Rearranging (16.26) yields:

$$ \frac{{-\mathrm{ d}T}}{{(W/\rho Q\mathrm{ Cp}+UA)+\left( {{T_{\mathrm{ a}}}-T} \right)}}=-\mathrm{ d}t\left( {\frac{Q}{V} + \frac{\;UA }{{\rho V\mathrm{ Cp}}}} \right). $$

(16.27)

Integrating (16.27):

$$ \begin{array}{lll}\ln \left. {\left\{ {\frac{W}{{\left( {\rho Q\mathrm{ Cp}+UA} \right)}}+\left( {{T_{\mathrm{ a}}}-T} \right)} \right.} \right\}=\left[ {-t\left( {\frac{Q}{V} + \frac{\;UA }{{\rho V\mathrm{ Cp}}}} \right)+\mathrm{ Constant}} \right].\end{array} $$

(16.28)

Let the initial temperature of the air in the room be T ₀. The boundary condition is then T = T ₀ at t = 0. Equation (16.28) then reduces to

$$ \begin{array}{llll}T-{T_{\mathrm{ a}}}=\frac{W}{{\left( {\rho Q\mathrm{ Cp}+UA} \right)}}\left[ {1 - \exp \left\{ {-t\left( {\frac{Q}{V}+\frac{\;UA }{{\rho V\mathrm{ Cp}}}} \right)} \right\}} \right]+({T_0}-{T_{\mathrm{ a}}})\exp \left\{ {-t\left( {\frac{Q}{V}+\frac{\;UA }{{\rho V\mathrm{ Cp}}}} \right)} \right\}.\end{array}$$

(16.29)

The number of air changes per second is n = Q/V. Hence (16.29) can be written:

$$ T-{T_{\mathrm{ a}}}=\frac{W}{{\left( {\rho nV\mathrm{ Cp}+UA} \right)}}\left[ {1 - \exp \left\{ {-t\left( {n+\frac{UA }{{\rho V\mathrm{ Cp}}}} \right)} \right\}} \right]+({T_0}-{T_{\mathrm{ a}}})\exp \left\{ {-t\left( {n+\frac{\;UA }{{\rho V\mathrm{ Cp}}}} \right)} \right\}. $$

(16.30)

which, for steady state, or as t tends to infinity, reduces to

$$ T-{T_{\mathrm{ a}}}=\frac{W}{{\left( {\rho Q\mathrm{ Cp} + UA} \right)}}=\frac{W}{{\left( {\rho nV\mathrm{ Cp} + UA} \right)}}. $$

(16.31)

Transient Humidity

Consider a room of volume V, with an air exchange rate of Q m³/s. A mass balance on the water vapor entering and leaving the room yields:

Rate of change of humidity in room = Humidity transported in/out + Amount generated

$$ V\mathrm{ d}H/\mathrm{ dt}=Q\left( {\mathrm{ Ha}-H} \right)+\mathrm{ Mp}/\rho, $$

(16.32)

where H and Ha are the humidities (kg H₂O/kg dry air) of the room and outside air, respectively, and Mp is the moisture production rate (kg H₂O/s). Rearranging (16.32) yields:

$$ \left( {-\mathrm{ d}H} \right)/\left( {\mathrm{ Ha}+\mathrm{ Mp}/Q\rho -H} \right)=\left( {-\mathrm{ d}t} \right)/VQ. $$

(16.33)

Integrating (16.33) with the boundary condition H = Ho at t = 0 and substituting n = Q/V yields:

$$ \left( {H-\mathrm{ Ho}} \right)=\left\{ {\mathrm{ Ha}-\mathrm{ Ho}+\mathrm{ Mp}/\rho nV} \right\}\left\{ {1-\exp \left( {-nt} \right)} \right\}. $$

(16.34)

Transient Pollutant

Consider a room of volume V, with an air exchange rate of Q m³/s. A mass balance on the pollutant entering and leaving the room yields:

$$ \begin{array}{lll}\mathrm{ Rate}\ \mathrm{ of}\ \mathrm{ change}\ \mathrm{ of}\ \mathrm{ pollutant}\ \mathrm{ concentration}\ \mathrm{ in}\ \mathrm{ room}\cr =\mathrm{ Pollutant}\ \mathrm{ convected}\ \mathrm{ in}/\mathrm{ out}+\mathrm{ Emissions}-\mathrm{ Removal}\end{array} $$

$$ V\frac{{\mathrm{ d}C}}{\mathrm{ d}\mathrm{ t}}=Q{C_{\mathrm{ a}}}+E-QC-kCV $$

(16.35)

where C and C _a are the pollutant concentrations (kg pollutant/kg dry air) of the room and outside air, respectively, E is the emission rate (kg/s), and k is the removal rate (1/s) of the pollutant in the room. Rearranging and integrating (16.35) with the boundary condition C = C _o at t = 0 yields:

$$ \begin{array}{llll} C =\left( {\frac{{Q{C_{\mathrm{ a}}}+E}}{Q+kV }} \right)\left\{ {1-\exp \left( {-\left( {\frac{Q}{V}+k} \right)t} \right)} \right\}+{C_{\mathrm{ o}}}\exp \left( {-\left( {\frac{Q}{V}+k} \right)t} \right) \cr =\left( {\frac{{n{C_{\mathrm{ a}}}+E/V}}{n+k }} \right)\left\{ {1-\exp \left( {-\left( {n+k} \right)t} \right)} \right\}+{C_{\mathrm{ o}}}\exp \left( {-\left( {n+k} \right)t} \right) \\\end{array}. $$

(16.36)

At steady state,

$$ C=\frac{{Q{C_{\mathrm{ a}}}+E}}{Q+kV }=\frac{{n{C_{\mathrm{ a}}}+E/V}}{n+k }. $$

(16.37)

Also, if k tends to zero and the air outside is “clean” (C _a = 0), then for C_o = 0

$$ C=\frac{E}{Q}\left\{ {1-\exp \left( {-\frac{Q}{V}t} \right)} \right\}=\frac{E}{nV}\left\{ {1-\exp \left( {-nt} \right)} \right\}. $$

(16.38)

Appendix 2: Humidity and Condensation

The terms humidity and relative humidity have specific definitions in psychrometry. These are:

Humidity, H, is the ratio of the mass of water vapor per unit mass of dry air,

and,

Relative Humidity, RH, is the ratio of partial vapor pressure, p _v to the saturation vapor pressure, p _vs, at the same temperature.

$$\begin{array}{lllll}H=\displaystyle\frac{{{M_{\mathrm{ water}}}{p_v}}}{{{M_{\mathrm{ air}}}(p-{p_{\mathrm{ v}}})}}=\frac{{18{p_{\mathrm{ v}}}}}{{29(p-{p_{\mathrm{ v}}})}}=0.622\displaystyle\frac{{{p_{\mathrm{ v}}}}}{{(p-{p_{\mathrm{ v}}})}} \hfill\\{\mathrm{ RH}=\displaystyle\frac{{{p_{\mathrm{ v}}}}}{{{p_{\mathrm{ v}\mathrm{ s}}}}}} \hfill \end{array}. $$

(16.39)

It follows from the perfect gas law that

$$ \begin{array}{llll} H=\displaystyle\frac{{{M_{\mathrm{ water}}}{p_{\mathrm{ v}}}}}{{{M_{\mathrm{ air}}}(p-{p_{\mathrm{ v}}})}}=\frac{{18{p_{\mathrm{ v}}}}}{{29(p-{p_{\mathrm{ v}}})}}=0.622\displaystyle\frac{{{p_{\mathrm{ v}}}}}{{(p-{p_{\mathrm{ v}}})}} \hfill \\{\mathrm{ RH}=\displaystyle\frac{{{p_{\mathrm{ v}}}}}{{{p_{\mathrm{ v}\mathrm{ s}}}}}} \hfill \\\end{array}, $$

(16.40)

where M _water and M _air are the molar masses of water and air respectively, and p, p _v and p _vs are the total pressure in the room, the water vapor pressure in the room and the water saturated vapor pressure at the room temperature.

Condensation will start to occur when the relative humidity reaches 100 %. The following procedure can be used to predict whether condensation is likely to occur within a ventilated room:

• 1. Evaluate H and T from the equations given in Appendix 1.

• 2. From the estimate of T find p _vs from steam tables.

• 3. From the estimate of H compute p _v.

• 4. Compare the vapor pressure values; if p _v > p _vs then condensation will occur.

Normally, condensation will occur on the coldest surfaces when the local RH value (this is the RH value computed using the temperature of the cold surface) reaches 100 %. In order to predict the likelihood of condensation on cold surfaces, the same procedure as that identified above can be adopted, using the temperature of the cold surface to estimate p _vs, rather than the temperature of the air. The p _v is still evaluated using the computed bulk air humidity value, H.