An apparatus for determining a flow rate of carbon dioxide, comprising: (i) a delivery line through which flows gaseous carbon dioxide, liquid carbon dioxide, or a combination of gaseous and liquid carbon dioxide; (ii) an orifice at the distal end of the delivery line, through which the carbon dioxide exits from the delivery line, wherein the diameter of the delivery line as it joins the orifice is D p and the diameter of the orifice is D o, and wherein D o is less than D p; (iii) a first temperature sensor proximal to the orifice and configured to detect a first temperature, T, of carbon dioxide in the delivery line and to transmit the detected first temperature to a flow rate calculation system; (iv) a pressure sensor proximal to the orifice and configured to detect a pressure, P u, of carbon dioxide in the delivery line and to transmit the detected pressure to the flow rate calculation system; (v) a second temperature sensor distal to the orifice and configured to detect a second temperature, T D, of carbon dioxide exiting the orifice and to transmit the detected second temperature to the flow rate calculation system; (vi) the flow rate calculation system, wherein the flow rate calculation system is configured to a. at a first time, determine from T D whether the carbon dioxide in the delivery line when it reaches the orifice is 100% gas or 100% liquid, or a mix of gas and liquid, and b. calculate an instantaneous flow rate for the first time, wherein 1. when the carbon dioxide in the delivery line as it is delivered to the orifice is 100% gas or 100% liquid, the flow rate is calculated from D p, D o, and P u, and T for the first time; and 2. when the carbon dioxide in the delivery line as it is delivered to the orifice is a mixture of gas and liquid, the flow rate is calculated from D o, P u, T, and T D at the first time. (i) a delivery line through which flows gaseous carbon dioxide, liquid carbon dioxide, or a combination of gaseous and liquid carbon dioxide; (ii) an orifice at the distal end of the delivery line, through which the carbon dioxide exits from the delivery line, wherein the diameter of the delivery line as it joins the orifice is D p and the diameter of the orifice is D o, and wherein D o is less than D p; (iii) a first temperature sensor proximal to the orifice and configured to detect a first temperature, T, of carbon dioxide in the delivery line and to transmit the detected first temperature to a flow rate calculation system; (iv) a pressure sensor proximal to the orifice and configured to detect a pressure, P u, of carbon dioxide in the delivery line and to transmit the detected pressure to the flow rate calculation system; (v) a second temperature sensor distal to the orifice and configured to detect a second temperature, T D, of carbon dioxide exiting the orifice and to transmit the detected second temperature to the flow rate calculation system; (vi) the flow rate calculation system, wherein the flow rate calculation system is configured to a. at a first time, determine from T D whether the carbon dioxide in the delivery line when it reaches the orifice is 100% gas or 100% liquid, or a mix of gas and liquid, and b. calculate an instantaneous flow rate for the first time, wherein 1. when the carbon dioxide in the delivery line as it is delivered to the orifice is 100% gas or 100% liquid, the flow rate is calculated from D p, D o, and P u, and T for the first time; and 2. when the carbon dioxide in the delivery line as it is delivered to the orifice is a mixture of gas and liquid, the flow rate is calculated from D o, P u, T, and T D at the first time. a. at a first time, determine from T D whether the carbon dioxide in the delivery line when it reaches the orifice is 100% gas or 100% liquid, or a mix of gas and liquid, and b. calculate an instantaneous flow rate for the first time, wherein 1. when the carbon dioxide in the delivery line as it is delivered to the orifice is 100% gas or 100% liquid, the flow rate is calculated from D p, D o, and P u, and T for the first time; and 2. when the carbon dioxide in the delivery line as it is delivered to the orifice is a mixture of gas and liquid, the flow rate is calculated from D o, P u, T, and T D at the first time. 1. when the carbon dioxide in the delivery line as it is delivered to the orifice is 100% gas or 100% liquid, the flow rate is calculated from D p, D o, and P u, and T for the first time; and 2. when the carbon dioxide in the delivery line as it is delivered to the orifice is a mixture of gas and liquid, the flow rate is calculated from D o, P u, T, and T D at the first time.